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Chapter 0: Problem 56

Simplify each exponential expression. $$\left(10 x^{2}\right)^{-3}$$

Short Answer

Expert verified
\((10x^{2})^{-3} = \frac{1}{1000x^{6}}\)

Step by step solution

01

Rewrite the expression

Rewrite the expression \((10x^{2})^{-3}\) as \(\frac{1} {(10x^{2})^{3}}\). This is because any number raised to a negative exponent becomes the denominator of a fraction with 1 as the numerator.
02

Apply the power rule

Now, apply the power rule which states that (a^m)^n = a^(m*n). The expression then will convert to \(\frac{1}{10^{3}(x^{2})^{3}}\) which simplifies further to \(\frac{1}{1000x^{6}}\).
03

Give the final answer

Thus, \((10x^{2})^{-3}\) simplifies to \(\frac{1}{1000x^{6}}\).

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Key Concepts

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

Negative Exponents
Understanding negative exponents is crucial for simplifying exponential expressions. A negative exponent indicates that the base should be moved to the denominator of a fraction. For example, if you have an expression like \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\). This idea helps transform complex expressions into simpler forms.

Consider the expression \((10x^{2})^{-3}\). The negative three indicates that both \(10\) and \(x^{2}\) should be in the denominator, resulting in:
  • \((10x^{2})^{-3} = \frac{1}{(10x^{2})^{3}}\)
Recognizing the role of negative exponents allows us to make expressions more manageable and is a foundational step in many algebraic problems.
Power Rule
The power rule in exponents is a useful tool for dealing with expressions in parentheses raised to another power. The rule states \(((a^m)^n = a^{m \cdot n})\). This allows you to multiply the exponents effectively.

When applying the power rule to \((10x^2)^3\) in our example:
  • The \(10\) becomes \(10^3\)
  • The \(x^2\) is raised to the third power, becoming \(x^{2 \cdot 3} = x^6\)
Therefore, \((10x^2)^3\) simplifies to \(10^3 \times x^6\). Understanding how to apply the power rule helps you simplify expressions by systematically combining powers, making calculations easier.
Algebraic Simplification
Algebraic simplification uses rules and properties to make expressions more concise. By applying concepts like negative exponents and the power rule, you can transform and reduce expressions efficiently.

In the expression \(\frac{1}{10^3 x^6}\), each part has been simplified from the negative exponent:
  • \(10^3 = 1000\), turning \(\frac{1}{10^3}\) into \(\frac{1}{1000}\)
  • \(x^6\) remains as is
Thus, \((10x^2)^{-3}\) simplifies elegantly to \(\frac{1}{1000x^6}\).

Through algebraic simplification, complex expressions become easier to work with and understand, an essential skill in mathematics.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{3}-64=(x+4)\left(x^{2}+4 x-16\right)$$

Explain how to determine which numbers must be excluded from the domain of a rational expression.

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$
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