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Chapter 0: Problem 22

Evaluate each exponential expression. $$\frac{3^{4}}{3^{7}}$$

Short Answer

Expert verified
The exponential expression \(\frac{3^{4}}{3^{7}}\) simplifies to \(\frac{1}{27}\)

Step by step solution

01

Identify the Base and the Exponents

Here, we notice that the base is 3 in both the numerator and the denominator. The exponent in the numerator is 4 and in the denominator is 7.
02

Apply the Laws of Exponents (Quotient Rule)

According to the quotient rule of exponents, when dividing like bases, you should subtract the exponent of the denominator from the exponent of the numerator: \(3^{4}/3^{7} = 3^{4-7} = 3^{-3}\)
03

Simplify the Negative Exponent

A negative exponent indicates that the expression is the reciprocal of the base raised to the absolute value of that exponent. Simplify the expression \(3^{-3}\) to \(\frac{1}{3^{3}}\)
04

Evaluate the Power

Evaluate \(3^{3}\) to find the simplification of the expression, which should be \(\frac{1}{3^{3}} = \frac{1}{27}\)

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Most popular questions from this chapter

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)

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