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Chapter 6: Problem 6

Use the properties of exponents to rewrite each expression without negative exponents. a. \(\left(2 x^{3}\right)^{2}\left(3 x^{4}\right)\) b. \(\left(5 x^{4}\right)^{0}\left(2 x^{2}\right)\) c. \(3(2 x)^{3}(3 x)^{-2}\) (a) d. \(\left(\frac{2 x^{4}}{3 x}\right)^{-3}\)

Short Answer

Expert verified
a. 12x^{10}; b. 2x^2; c. \frac{8}{3}x; d. \frac{27}{8x^9}.

Step by step solution

01

Simplify \\( (2x^3)^2(3x^4) \\\)

Apply the power rule \( (a^m)^n = a^{mn} \) to \( (2x^3)^2 \). This gives \( 2^2(x^3)^2 = 4x^6 \). Now multiply by \( 3x^4 \). This results in \( 4x^6 \cdot 3x^4 = 12x^{6+4} = 12x^{10} \).
02

Simplify \\( (5x^4)^0(2x^2) \\\)

Using the exponent zero rule \( a^0 = 1 \) for any non-zero \( a \), \( (5x^4)^0 = 1 \). So the expression becomes \( 1 \cdot 2x^2 = 2x^2 \).
03

Simplify \\( 3(2x)^3(3x)^{-2} \\\)

Start by using the power rule on \( (2x)^3 \): \( 2^3x^3 = 8x^3 \). Then, use the negative exponent rule \( a^{-n} = \frac{1}{a^n} \) for \( (3x)^{-2} \), which gives \( \frac{1}{9x^2} \). The expression now becomes \( 3 \cdot 8x^3 \cdot \frac{1}{9x^2} = \frac{24x^3}{9x^2} \). Simplifying gives \( \frac{24}{9}x^{3-2} = \frac{8}{3}x \).
04

Simplify \\( \left(\frac{2x^4}{3x}\right)^{-3} \\\)

Use the negative exponent rule \( a^{-n} = \frac{1}{a^n} \) to convert to positive exponents: \( \left(\frac{3x}{2x^4}\right)^{3} \). Distribute the exponent: \( \frac{3^3x^3}{2^3(x^4)^3} = \frac{27x^3}{8x^{12}} \). Simplify by subtracting exponents of \( x \) (3 from 12): \( \frac{27}{8x^9} \).

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Key Concepts

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

Power Rule
One of the fundamental rules of exponents is the power rule. This rule states that when you raise an exponent to another exponent, you multiply the exponents together. Mathematically, it is expressed as \((a^m)^n = a^{mn}\).
For example, in solving the expression \((2x^3)^2\), we apply the power rule. We square both the base and the exponent, resulting in \(2^2(x^3)^2\), simplifying to \(4x^6\).
Using the power rule helps streamline simplification of expressions and is particularly useful in working with polynomial equations. It鈥檚 like expanding your equation into a more manageable form.
Negative Exponents
Negative exponents can be tricky, but they follow a simple rule: \(a^{-n} = \frac{1}{a^n}\). This means that a negative exponent represents a reciprocal. Suppose you have a negative exponent; you can convert it into a fraction, flipping the base into the denominator and making the exponent positive.
For instance, with the term \((3x)^{-2}\), using the negative exponent rule transforms this into \(\frac{1}{9x^2}\).
Negative exponents allow you to move components of an expression across the fraction line, turning division into multiplication and providing another crucial strategy for manipulating exponential expressions.
Zero Exponent Rule
The zero exponent rule is straightforward: any non-zero number raised to the power of zero is equal to one. In notation, this is \(a^0 = 1\).
This simplifies parts of expressions that might seem complex. For example, look at \((5x^4)^0\). Regardless of the base, since the exponent is zero, this expression equates to 1.
The zero exponent rule can significantly simplify expressions. It's a handy tool for eliminating entire sections of your equation, reducing it down to core components without affecting other variables present in your expression.
Simplifying Expressions
Simplifying expressions involves using rules like the power rule, negative exponents, and zero exponent rule, often in combination. The goal is to reduce complexity by consolidating like terms and minimizing the degree of expression.
Consider the expression \( \frac{24x^3}{9x^2} \). By simplifying, we break it down as \( \frac{24}{9}x^{3-2} = \frac{8}{3}x \).
Simplification transforms convoluted expressions into more comprehensible forms, making mathematics less daunting and allowing for cleaner, easier results. The key is to patiently apply these rules iteratively to resolve and clarify each part of the expression.

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Most popular questions from this chapter

Rewrite each expression with a single exponent. a. \(\left(3^{5}\right)^{8}\) b. \(\left(7^{3}\right)^{4}\) c. \(\left(x^{6}\right)^{2}\) d. \(\left(y^{8}\right)^{5}\)

Use the properties of exponents to rewrite each expression with only positive exponents. a. \(4 x^{3} \cdot\left(3 x^{5}\right)^{3}\) b. \(\frac{60 x^{4} y^{4}}{15 x^{3} y}\) c. \(3^{2} \cdot 2^{3}\) d. \(\frac{\left(8 x^{3}\right)^{2}}{\left(4 x^{2}\right)^{3}}\) (d) e. \(x^{-3} y^{4}\) f. \((2 x)^{-3}\) g. \(2 x^{-3}\) h. \(\frac{2 x^{-1}}{\left(3 y^{2}\right)^{-3}}\)

APPLICATION Camila received a \(\$ 1,200\) prize for one of her essays. She decides to invest \(\$ 1,000\) of it for college. Her bank offers two options. The first is a regular savings account that pays \(2.5 \%\) interest every 6 months. The second is a certificate of deposit that pays \(5 \%\) interest each year. a. With the savings account, how much would Camila have after 1 year? After 2 years? (a) b. With the certificate of deposit, how much would Camila have after 1 year? After 2 years? c. Explain why you get different results for \(12 \mathrm{a}\) and \(\mathrm{b}\). (i)

The population of a town is currently 45,647 . It has been growing at a rate of about \(2.8 \%\) per year. a. Write an expression in the form \(45,647(1+0.028)^{x}\) for the current population. b. What does the expression \(45,647(1+0.028)^{-12}\) represent in this situation? (a) c. Write and evaluate an expression for the population 8 years ago. (A) d. Write expressions without negative exponents that are equivalent to the exponential expressions from \(4 \mathrm{~b}\) and \(\mathrm{c}\). (a)

APPLICATION Tamara works at a bookstore, where she earns \(\$ 7.50\) per hour. a. Her employer is pleased with her work and gives her a \(3.5 \%\) raise. What is her new hourly rate? b. A few weeks later business drops off dramatically. The employer must reduce wages. He decreases Tamara's latest wage by \(3.5 \%\). What is her hourly rate now? c. What is the final result of the two pay changes? Explain. (H)
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