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Chapter 6: Problem 20

Simplify. Assume that no variable equals 0. $$ \left(z^{2}\right)^{5} $$

Short Answer

Expert verified
The simplified form is \(z^{10}\).

Step by step solution

01

Identify the Power Rule for Exponents

When you have an expression of the form \(a^m\)^n, you multiply the exponents: this is the power rule for exponents. Using this rule, \(a^{m imes n}\) becomes the simplified form.
02

Apply the Power Rule to Simplify the Expression

For the given expression \(\left(z^2\right)^5\), identify the base \(z\) with its exponent \(2\). Multiply this exponent by the outer exponent \(5\) to get the new exponent. This results in \(z^{2 \times 5} = z^{10}\).

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Key Concepts

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

Exponents
Exponents, also known as powers, are a shorthand way of expressing repeated multiplication of the same number. If you see something like \(a^b\), it means that the base \(a\) is multiplied by itself \(b\) times. This concept is fundamental in mathematics as it helps in simplifying expressions and solving equations.
For example, \(2^3 = 2 \times 2 \times 2 = 8\). In this example:
  • \(2\) is the base
  • \(3\) is the exponent
  • \(8\) is the result of the multiplication
Using exponents makes calculations quicker and more efficient. It allows us to express very large or very small numbers in a more concise form, which is especially useful in scientific notation.
Simplifying Expressions
Simplifying expressions is about making them as clean and efficient as possible, by reducing them to their simplest form. This often means combining like terms or applying mathematical rules like the power rule for exponents. Simplifying can make expressions easier to work with and solve.
Take the example from the original exercise: \(\left(z^2\right)^5\). By applying the power rule for exponents, you simplify it to \(z^{10}\). This makes the expression easier to handle in further mathematical operations.
  • Combining like terms
  • Using rules, such as the power rule, to reduce the complexity
  • Makes calculations more straightforward and manageable
Remember, the goal of simplifying is to make expressions look neater without changing their value. This is achieved by following consistent mathematical procedures.
Multiplying Exponents
When you come across a situation where you need to multiply exponents, like \((a^m)^n\), it's important to apply the correct rules to simplify it correctly. This is where the power rule for exponents comes into play.
The power rule states that to simplify an expression of the form \((a^m)^n\), you multiply the exponents: \(a^{m \times n}\). Let's break it down using the exercise we have:
  • The base is \(z\), which is being raised to the power \(2\) inside the parenthesis.
  • This entire expression is then raised to the power of \(5\).
  • According to the power rule, you multiply the exponents: \(z^{2 \times 5} = z^{10}\).
This powerful rule simplifies complex exponent expressions and helps perform calculations in a more streamlined and error-free manner.

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Most popular questions from this chapter

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 2 x^{2}+3 x+2=0 $$

Use synthetic substitution to show that \(x-8\) is a factor of \(x^{3}-4 x^{2}-29 x-24 .\) Then find any remaining factors.

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{2}-8 x+6 $$

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Use synthetic substitution to estimate the sales for 2008 .

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=x^{3}+x^{2}-7 x-3 $$
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