Exponents
56 flashcards covering Exponents for the GMAT Quantitative section.
Exponents are a mathematical tool that represents repeated multiplication in a compact form. For example, if you see 2^4, it means multiplying 2 by itself four times: 2 × 2 × 2 × 2 = 16. This concept is essential for handling large numbers, patterns in sequences, and algebraic expressions, making it a building block for more advanced math topics like equations and functions.
On the GMAT Quantitative section, exponents show up in problems involving algebraic manipulation, such as simplifying expressions or solving equations with powers. Common question types include applying rules like the product rule (a^m × a^n = a^(m+n)) or dealing with fractional and negative exponents. Watch out for traps like confusing exponent orders or misapplying rules with roots. Focus on mastering these properties to handle time-pressured questions accurately.
Practice exponent rules with sample problems to spot errors quickly.
Terms (56)
- 01
Exponent
An exponent is a superscript number in an expression like a^b, indicating that the base a is multiplied by itself b times.
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Base
In an expression like a^b, the base is the number a that is being raised to the power b, representing the factor being multiplied.
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Positive Integer Exponent
A positive integer exponent means multiplying the base by itself that many times, such as 2^3 equaling 2 multiplied by 2 multiplied by 2.
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Negative Exponent
A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent, so a^{-n} equals 1 divided by a^n.
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Zero Exponent
Any non-zero number raised to the power of zero equals 1, as in a^0 = 1 for a not equal to zero.
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Product Rule for Exponents
When multiplying two expressions with the same base, add the exponents: for bases a, a^m times a^n equals a raised to the power of m plus n.
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Quotient Rule for Exponents
When dividing two expressions with the same base, subtract the exponents: for bases a, a^m divided by a^n equals a raised to the power of m minus n.
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Power Rule for Exponents
When raising a power to another power, multiply the exponents: for an expression like (a^m)^n, it equals a raised to the power of m times n.
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Product to a Power Rule
When raising a product to a power, apply the exponent to each factor: (a b)^n equals a^n times b^n.
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Quotient to a Power Rule
When raising a quotient to a power, apply the exponent to both the numerator and the denominator: (a / b)^n equals a^n divided by b^n.
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Adding Exponents with Same Base
Exponents with the same base can only be added when multiplying the expressions, as in a^m times a^n equals a^(m+n), but not for addition alone.
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Subtracting Exponents with Same Base
Exponents with the same base can be subtracted when dividing, as in a^m divided by a^n equals a^(m-n), provided m is greater than n.
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Multiplying Exponents with Different Bases
When multiplying expressions with different bases but the same exponent, multiply the bases and keep the exponent, like a^n times b^n equals (a b)^n.
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Dividing Exponents with Different Bases
When dividing expressions with different bases but the same exponent, divide the bases and keep the exponent, like a^n divided by b^n equals (a / b)^n.
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Fractional Exponent
A fractional exponent represents a root and a power, where a^(m/n) means the n-th root of a raised to the m power, such as a^(1/2) being the square root of a.
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Square Root as Exponent
The square root of a number a can be written as a raised to the power of 1/2, meaning a^(1/2) equals the number that, when multiplied by itself, gives a.
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Cube Root as Exponent
The cube root of a number a can be expressed as a raised to the power of 1/3, meaning a^(1/3) equals the number that, when multiplied by itself three times, gives a.
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Rational Exponent
A rational exponent is a fraction that indicates both a root and a power, where a^(p/q) equals the q-th root of a raised to p, and it must be simplified carefully.
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Negative Fractional Exponent
A negative fractional exponent means taking the reciprocal and then applying the root and power, so a^(-m/n) equals 1 divided by the n-th root of a raised to m.
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Simplifying Expressions with Exponents
Simplifying expressions with exponents involves applying the rules to combine like terms, eliminate negative exponents, and ensure all exponents are positive and in lowest terms.
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Exponents in Scientific Notation
Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10, like 5.6 times 10^3, which is useful for very large or small numbers.
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Converting to Scientific Notation
To convert a number to scientific notation, move the decimal point so it's after the first non-zero digit and multiply by 10 raised to the number of places moved, adjusting for direction.
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Common Trap: Distributing Exponents
A common error is incorrectly distributing exponents over addition, as (a + b)^n does not equal a^n + b^n, but rather requires expanding using the binomial theorem.
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Common Trap: Negative Bases
With negative bases, even exponents result in positive values while odd exponents keep the sign, so care must be taken with roots and fractions to avoid imaginary numbers.
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Order of Operations with Exponents
In expressions with multiple operations, exponents are evaluated before multiplication and division, but after parentheses, following the PEMDAS rule.
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Parentheses with Exponents
Parentheses indicate that the exponent applies to the entire expression inside, so (ab)^n means both a and b are raised to n, unlike a b^n without parentheses.
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Expressing with Positive Exponents
To express a term with positive exponents, rewrite negative exponents as fractions, such as a^{-n} as 1/a^n, and simplify accordingly.
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Reducing Fractions with Exponents
When reducing fractions with exponents, subtract exponents in the numerator and denominator if bases are the same, or factor and cancel common terms.
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Solving Exponential Equations
To solve equations where variables are in the exponent, such as a^x = b, take the logarithm of both sides or recognize when bases are the same to set exponents equal.
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Exponential Equations with Same Base
If two sides of an equation have the same base, set the exponents equal, like if a^x = a^y, then x equals y, assuming a is positive and not one.
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Strategy for Simplifying Complex Exponents
Break down complex expressions by applying exponent rules step by step, starting with the innermost parentheses and combining like bases before simplifying.
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Exponents and Roots Together
Expressions with both exponents and roots can be simplified by converting roots to fractional exponents and then applying the rules, such as simplifying the square root of a^4.
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Multiplying Different Roots
When multiplying roots of the same index, combine them under a single root, like the square root of a times the square root of b equals the square root of a times b.
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Dividing Different Roots
When dividing roots of the same index, combine them as a single root of the quotient, like the square root of a divided by the square root of b equals the square root of a divided by b.
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Rationalizing Denominators with Exponents
To rationalize a denominator with a root, multiply numerator and denominator by the appropriate power to eliminate the root, such as multiplying by the conjugate for square roots.
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Comparing Expressions with Exponents
To compare two expressions with exponents, express them with the same base or exponent, or evaluate numerically if possible, to determine which is larger.
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Exponents in Word Problems
In word problems, exponents often represent growth or decay, such as compound interest, where the formula involves raising a base to a power over time.
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Compound Interest Formula
The compound interest formula is A = P(1 + r/n)^(nt), where exponents show how the principal grows over time with periodic compounding.
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Exponential Growth
Exponential growth means a quantity increases by a fixed percentage over equal intervals, modeled by y = a b^x, where b > 1.
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Exponential Decay
Exponential decay means a quantity decreases by a fixed percentage over equal intervals, modeled by y = a b^x, where 0 < b < 1.
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Logarithms as Inverse of Exponents
Logarithms are the inverse operation of exponents, so if y = a^x, then log base a of y equals x, useful for solving exponential equations.
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Common Logarithm
The common logarithm is base 10, written as log(y), and equals the exponent to which 10 must be raised to get y.
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Natural Logarithm
The natural logarithm is base e (approximately 2.718), written as ln(y), and equals the exponent to which e must be raised to get y.
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Properties of Logarithms with Exponents
Logarithms of powers use the property that log(a^b) equals b times log(a), which simplifies expressions involving exponents.
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Exponents in Inequalities
When dealing with inequalities and exponents, remember that if the base is positive and greater than 1, the inequality direction remains the same when taking roots or logs.
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Fractional Exponents in Equations
In equations with fractional exponents, raise both sides to the reciprocal of the fraction to eliminate the root, then solve for the variable.
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Even and Odd Exponents
Even exponents always result in a positive value for real numbers, while odd exponents preserve the sign of the base.
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Absolute Value with Exponents
When dealing with roots of negative numbers, absolute value may be needed, as the principal root is positive, like the square root of 9 equals 3, not -3.
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Nested Exponents
Nested exponents, like a^(b^c), are evaluated from right to left, so a^(b^c) means a raised to the power of (b^c), not (a^b)^c.
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Exponents and Zero Base
A base of zero raised to a positive exponent is zero, but zero raised to zero or a negative exponent is undefined.
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Exponents Greater Than 1
When the base is greater than 1, increasing the exponent increases the value, which is key in growth problems.
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Exponents Between 0 and 1
When the base is between 0 and 1, increasing the exponent decreases the value, as in decay scenarios.
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Strategy for Factoring with Exponents
To factor expressions with exponents, look for common factors or use difference of powers formulas, like a^2 - b^2 = (a - b)(a + b).
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Difference of Powers
The difference of squares is a^2 - b^2 = (a - b)(a + b), and higher powers like difference of cubes follow patterns for factoring.
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Sum of Powers
Sums of powers, like a^2 + b^2, do not factor nicely over the reals, unlike differences, so recognize when factoring is possible.
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Exponents in Polynomials
Exponents define the degree of terms in polynomials, with the highest exponent determining the polynomial's degree.