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Chapter 2: Problem 4

Evaluate the given expression. Do not use a calculator. $$ \frac{2^{-6}}{6^{-2}} $$

Short Answer

Expert verified
The short answer based on the step-by-step solution is: \(\frac{2^{-6}}{6^{-2}} = \frac{9}{16}\).

Step by step solution

01

Write the given expression as a single fraction

Let's write the given expression as a single fraction: \[ \frac{2^{-6}}{6^{-2}} = 2^{-6} \cdot \frac{1}{6^{-2}} \]
02

Use exponent properties to simplify the expression

Since we have a negative exponent, we can rewrite the expression as: \[ 2^{-6} \cdot \frac{1}{6^{-2}} = \frac{1}{2^6} \cdot \frac{1}{\frac{1}{6^2}} \] Now, we can use the identity \(\frac{1}{\frac{1}{a}} = a\) to further simplify: \[ \frac{1}{2^6} \cdot \frac{1}{\frac{1}{6^2}} = \frac{1}{2^6} \cdot 6^2 \]
03

Combine fractions under a common denominator

To combine the above expression into a single fraction, we will use the identity \(\frac{1}{a} \cdot b = \frac{b}{a}\): \[ \frac{1}{2^6} \cdot 6^2 = \frac{6^2}{2^6} \]
04

Rewrite the expression with base 2 exponents

Express the numerator 6^2 in terms of base 2. Since we know that 6 is 2 times 3, we can rewrite \(6^2\) as \((2 \cdot 3)^2\): \[ \frac{(2 \cdot 3)^2}{2^6} = \frac{(2^2 \cdot 3^2)}{2^6} \]
05

Simplify the expression using the exponent properties

Using the exponent properties, we can simplify the expression as: \[ \frac{(2^2 \cdot 3^2)}{2^6} = \frac{2^2}{2^6} \cdot 3^2 = \frac{2^{2-6}}{1} \cdot 3^2 = 2^{-4} \cdot 3^2 \]
06

Calculate the final result

Calculate the value of the simplified expression: \[ 2^{-4} \cdot 3^2 = \frac{1}{2^4} \cdot 3^2 = \frac{1}{16} \cdot 9 = \frac{9}{16} \] The final result is: \[ \frac{2^{-6}}{6^{-2}} = \frac{9}{16} \]

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Most popular questions from this chapter

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$

Find a number \(b\) such that 3 is a zero of the polynomial \(p\) defined by $$ p(x)=1-4 x+b x^{2}+2 x^{3} $$.

Suppose \(p\) is a polynomial and \(t\) is a number. Explain why there is a polynomial \(G\) such that $$ \frac{p(x)-p(t)}{x-t}=G(x) $$ for every number \(x \neq t\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r \circ s)(x) $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r-s)(x) $$
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