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Chapter 0: Problem 3

Evaluate each expression. $$ \left(2^{3}\right)^{2} $$

Short Answer

Expert verified
The expression \((2^3)^2\) evaluates to 64.

Step by step solution

01

Understand the Expression

The expression \((2^3)^2\) is a power raised to another power. It's essential to recognize this is an exponentiation problem.
02

Apply the Power of a Power Rule

The rule for a power raised to a power is to multiply the exponents. So, for \((a^m)^n\), it simplifies to \(a^{m \cdot n}\).
03

Simplify the Exponent

Apply the rule from Step 2: \((2^3)^2 = 2^{3\cdot2}\). This simplifies to \(2^6\).
04

Calculate the Final Value

Now we need to calculate \(2^6\). This means multiplying 2 by itself 6 times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2\) which equals 64.

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Key Concepts

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

Understanding the Power of a Power Rule
When we encounter an expression like \((a^m)^n\), it's easy to feel overwhelmed if not familiar with the power of a power rule. This mathematical rule is a shortcut to simplify expressions where an exponentiated number is raised to another exponent. Instead of expanding everything, use this rule to multiply the exponents. For example, in the expression \((2^3)^2\), you apply the power of a power rule to get \(2^{3 \times 2}\). This simplifies the problem significantly and turns \((2^3)^2\) into a simpler form \(2^6\).

Here are the key points about the power of a power rule:
  • It simplifies exponentiation problems by reducing steps.
  • You only need to multiply the exponents while keeping the base the same.
  • This rule applies universally to any base and set of exponents.
Techniques for Simplifying Exponents
Simplifying exponents involves reducing complex exponential expressions into simpler ones. When using the power of a power rule, you need to carefully follow each step to ensure accuracy. For instance, in our expression \((2^3)^2\), upon applying the rule, we computed \(2^{3 \cdot 2}\), which simplifies to \(2^6\).

Here鈥檚 how to effectively simplify exponents:
  • Identify all operations involving exponents in the expression.
  • Use appropriate exponent rules, such as power of a power, product rule, and quotient rule.
  • Simplify in stages to keep things manageable and avoid errors.
Approaching problems methodically with these techniques streamlines the process and makes dealing with exponents hassle-free.
Evaluating Expressions with Exponents
Once you've simplified an expression involving exponents, the final step is to evaluate it to find the numerical value. After applying the power of a power rule and simplifying to \(2^6\), what remains is the multiplication of 2, six times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Evaluating such expressions involves straightforward arithmetic.
  • Compute the repeated multiplication to arrive at the final number.
  • In our example, multiplying out gives 64.
Evaluation is the step where the abstract becomes concrete, offering a tangible result to verify the simplification and rule application were done correctly.

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Most popular questions from this chapter

Write each number in decimal notation. $$ 6 \times 10^{12} $$

\(77-82\) me Rationalize the denominator. $$ \frac{y}{\sqrt{3}+\sqrt{y}} $$

Easy Powers That Look Hard Calculate these expressions in your head. Use the Laws of Exponents to help you. $$ \begin{array}{ll}{\text { (a) } \frac{18^{5}}{9^{5}}} & {\text { (b) } 20^{6} \cdot(0.5)^{6}}\end{array} $$

Let \(a, b,\) and \(c\) be real numbers with \(a > 0, b < 0,\) and \( c < 0 .\) Determine the sign of each expression. \(\begin{array}{ll}{\text { (a) } b^{5}} & {\text { (b) } b^{10} \quad \text { (c) } a b^{2} c^{3}} \\ {\text { (d) }(b-a)^{3}} & {\text { (e) }(b-a)^{4}} \quad {\text { (f) } \frac{a^{3} c^{3}}{b^{6} c^{6}}}\end{array}\)

Distances between Powers \(\quad\) Which pair of numbers is closer together? $$ 10^{10} \text { and } 10^{50} \quad \text { or } \quad 10^{100} \text { and } 10^{101} $$
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