/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 27

Simplify each expression. $$ (3 z)^{2}\left(6 z^{2}\right)^{-3} $$

Short Answer

Expert verified
The expression simplifies to \(\frac{1}{24z^4}\).

Step by step solution

01

Simplify individual terms

First, simplify the expression \((3z)^2\) which becomes \(3^2 imes z^2 = 9z^2\).Next, simplify \((6z^2)^{-3}\). The exponent \(-3\) means take the reciprocal and cube the terms:\((6z^2)^{-3} = \left(\frac{1}{6z^2}\right)^3 = \frac{1}{6^3z^6} = \frac{1}{216z^6}\).
02

Multiply the simplified terms

To multiply the expressions \(9z^2\) and\(\frac{1}{216z^6}\), use the property of exponents \(z^a \cdot z^b = z^{a+b}\):\(9z^2 \times \frac{1}{216z^6} = \frac{9z^2}{216z^6}\).Simplify the fraction: \(\frac{9}{216} = \frac{1}{24}\).For the exponents, subtract: \(z^2 / z^6 = z^{2-6} = z^{-4}\).
03

Write the final expression

The simplified expression is \(\frac{1}{24z^4}\). This result is achieved by combining our fraction result with the simplified exponent: \(\frac{1}{24}\) multiplied by \(z^{-4} = \frac{1}{z^4}\).Thus, the entire expression becomes \(\frac{1}{24z^4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Understanding exponents is essential in simplifying expressions. An exponent refers to the number of times a number (the base) is multiplied by itself. For example,
  • \(3^2\) means \(3 \times 3\), which equals 9.
  • \(z^2\) implies \(z \times z\).
When dealing with expressions like \((3z)^{2}\), you apply the exponent to both the base number and the variable inside the parentheses. So \((3z)^{2}\) becomes \(3^2 \times z^2\), resulting in \(9z^2\).

Another concept to consider is the negative exponent. A negative exponent indicates dividing by that factor instead of multiplying. For example, \((6z^2)^{-3}\) requires using the reciprocal (as explained later) and taking the exponent of 3 for the denominator components.
Reciprocal
The reciprocal of a number is simply one divided by that number. It鈥檚 like flipping the number over. For example, the reciprocal of \(5\) is \(\frac{1}{5}\).
  • If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
  • Realize that the reciprocal of a whole number \(x\) is \(\frac{1}{x}\).

In the expression \((6z^2)^{-3}\), the negative exponent means you need to take the reciprocal of \(6z^2\). This becomes \(\left(\frac{1}{6z^2}\right)^3\), turning the negative exponent into a positive one by making the base a fraction.
Understanding reciprocals is crucial when working with negative exponents as it helps in rewriting the expression in a form that is easier to simplify.
Multiplying Fractions
To multiply fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example,
  • \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
In the solution, after turning \((6z^2)^{-3}\) into \(\frac{1}{216z^6}\), we have to multiply it by \(9z^2\), written as \(\frac{9z^2}{1}\), yielding:
  • \(\frac{9z^2}{1} \times \frac{1}{216z^6} = \frac{9z^2}{216z^6}\).
After multiplication, simplify the fraction.

Start with the coefficients: \(\frac{9}{216}\) simplifies to \(\frac{1}{24}\).Then subtract the exponents of \(z\): \(z^{2-6} = z^{-4}\), leaving the final expression in the simplified form of \(\frac{1}{24z^4}\). Working through each step with care leads to a clear and concise result.

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Most popular questions from this chapter

\(55-64=\) Simplify the compound fractional expression. $$ \frac{x^{-1}+y^{-1}}{(x+y)^{-1}} $$

\(55-64=\) Simplify the compound fractional expression. $$ \frac{\frac{a-b}{a}-\frac{a+b}{b}}{\frac{a-b}{b}+\frac{a+b}{a}} $$

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{3(x+2)^{2}(x-3)^{2}-(x+2)^{3}(2)(x-3)}{(x-3)^{4}} $$

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of significant digits indicated by the given data. $$ \frac{\left(3.542 \times 10^{-6}\right)^{9}}{\left(5.05 \times 10^{4}\right)^{12}} $$

Write each number in decimal notation. $$ 8.55 \times 10^{-3} $$
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