Evaluate each exponential expression: (33)/(36)

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 27

1장, 문제 27

Evaluate each exponential expression: (3³)/(3⁶)

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Recall the property of exponents that states when dividing powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Identify the base and the exponents in the expression \(\frac{3^3}{3^6}\). Here, the base is 3, the numerator exponent is 3, and the denominator exponent is 6.

Apply the exponent subtraction rule: \(\frac{3^3}{3^6} = 3^{3-6}\).

Simplify the exponent by performing the subtraction: \(3^{3-6} = 3^{-3}\).

Recognize that a negative exponent means the reciprocal: \(3^{-3} = \frac{1}{3^3}\). This is the simplified form of the expression.

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Properties of Exponents

Exponents follow specific rules that simplify expressions, such as when dividing powers with the same base, you subtract the exponents. For example, (a^m)/(a^n) = a^(m-n). This property helps in reducing complex exponential expressions efficiently.

Base Consistency in Exponents

When working with exponential expressions, the base must be the same to apply exponent rules like multiplication or division. In the given problem, both numerator and denominator have base 3, allowing the use of exponent subtraction.

Evaluating Powers

After simplifying the exponent expression, you may need to calculate the numerical value by raising the base to the resulting exponent. For example, 3^(3-6) = 3^(-3), which equals 1/(3^3) = 1/27.