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Chapter 5: Problem 36

Simplify each expression. Write each result using positive exponents only. See Examples I through 4. $$ \left(z^{5} x^{5}\right)^{-3} $$

Short Answer

Expert verified
\(\frac{1}{z^{15} x^{15}}\)

Step by step solution

01

Apply the Power of a Power Rule

When you have an expression of the form \((a^m)^n\), you can simplify it by multiplying the exponents according to the rule \(a^{m imes n}\). Apply this rule to each factor in the expression: \[ \left(z^{5} x^{5}\right)^{-3} = z^{5 \times -3} x^{5 \times -3}. \] This simplifies to \(z^{-15} x^{-15}\).
02

Convert Negative Exponents to Positive Exponents

The rule \(a^{-m} = \frac{1}{a^m}\) allows us to convert negative exponents to positive exponents. Apply this rule to both \(z^{-15}\) and \(x^{-15}\): \[ z^{-15} x^{-15} = \frac{1}{z^{15}} \cdot \frac{1}{x^{15}}. \] This expression is equivalent to \(\frac{1}{z^{15} x^{15}}\).

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Key Concepts

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

Exponents
Exponents are a shorthand notation for expressing repeated multiplication of a number by itself. When you have a base being multiplied by itself a certain number of times, we use an exponent to simplify the expression.
For example, instead of writing \( x \times x \times x \times x \times x \), you can write \( x^5 \). This shows that \( x \) is being multiplied by itself 5 times. The base \( x \) is the number being multiplied, and the number 5 is the exponent, indicating how many times the base is used as a factor.
Understanding exponents can greatly simplify working with large numbers or expressions in algebra, making calculations more manageable and less error-prone.
Negative Exponents
Negative exponents indicate that the base is on the wrong side of a fraction bar, or in other words, it needs to be turned into a reciprocal.
When you see an expression like \( a^{-n} \), it can be rewritten as \( \frac{1}{a^n} \). This means the base \( a \) now appears as the denominator in a fraction. This operation changes a potentially complex expression into something more understandable.
Let's use the example from the exercise. We had \( z^{-15} \) and \( x^{-15} \). Applying the rule, these became \( \frac{1}{z^{15}} \) and \( \frac{1}{x^{15}} \). Understanding this process helps simplify expressions using negative exponents to their positive forms.
Power of a Power Rule
The power of a power rule is a useful shortcut when dealing with expressions where you raise a power to another power.
The rule states that when you have something like \( (a^m)^n \), you simply multiply the exponents together to simplify it to \( a^{m \times n} \). This is because you're taking the entire power expression and repeatedly multiplying it by itself.
In the given exercise, \( (z^5 x^5)^{-3} \) used this rule. We multiplied \( 5 \times -3 \) for both \( z \) and \( x \), converting it to \( z^{-15} x^{-15} \).
Mastering this rule can help speed up solving algebraic expressions and reduce confusion when dealing with complex exponential forms.

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Most popular questions from this chapter

Multiply. See Example 7. $$ (4 x+5)(4 x-5) $$

Divide If the divisor contains 2 or more terms, use long division. $$ \frac{5-6 x^{2}}{x-2} $$

The area of the top of the Ping-Pong table is \(\left(49 x^{2}+70 x\right.\) \(-200\) ) square inches. If its length is \((7 x+20)\) inches, find its width.

Multiply each expression. $$ 2 x\left(x^{2}+7 x-5\right) $$

Simplify the following. See Section \(5.1 .\) $$ \frac{15 z^{4} y^{3}}{21 z y} $$
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