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Chapter 0: Problem 8

Simplify each expression. Leave answers with exponents. $$\left(6^{4}\right)^{3}$$

Short Answer

Expert verified
\(6^{12}\)

Step by step solution

01

Understanding the Power Rule

When an exponent is raised to another exponent, you multiply the exponents together. This rule can be expressed as \((a^m)^n = a^{m \cdot n}\). In our problem, this translates to \((6^4)^3\).
02

Apply the Power Rule

Use the power rule by multiplying the exponents: \(4 \times 3 = 12\). Thus, \((6^4)^3\) becomes \(6^{12}\).
03

Write the Simplified Expression

The simplified expression using the power rule is \(6^{12}\). This is the final answer and should be left as is with the exponent, as per the instructions.

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Key Concepts

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

Exponents
Exponents are a convenient way to express repeated multiplication of a number by itself. When you see a number written as a base with an exponent, like \( 6^4 \), it means you multiply the base number (6) by itself as many times as the exponent (4) indicates. So in this case, \( 6^4 \) equals \( 6 \times 6 \times 6 \times 6 \). Exponents help in simplifying expressions and dealing with very large numbers more easily.
  • Base: The number that gets multiplied.
  • Exponent: The number that tells how many times the base is used in a multiplication.
Remember, when you have to deal with an expression like \((6^4)^3\), it鈥檚 useful to apply the power rule, which involves multiplication of exponents to simplify the expression.
Simplifying Expressions
Simplifying expressions means making them easier to understand or work with. When simplifying, our goal is to express the mathematical statement in its simplest form without changing its value.
To simplify expressions involving exponents, you use rules like the power rule. In our example, we started with \((6^4)^3\). Using the power rule, where you multiply the exponent of the first power with the second power鈥檚 exponent, we found that it simplifies to \(6^{12}\). This reduces the complexity of the expression, making further calculations or comparisons much easier.
  • Identify like terms and operations.
  • Use rules like power and product rules effectively.
  • Simplify step-by-step to avoid errors.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In algebra, expressions can include exponents, which are pivotal in expressing repetitive multiplication scenarios neatly and concisely.
Algebraic expressions require a systematic approach to simplify and solve, particularly when they involve multiple operations and power rules.
  • Components: They include terms, coefficients, and possibly variables with exponents.
  • Manipulation: Often requires applying rules for operations such as the distributive property, combining like terms, and using exponent rules like the power rule.
Effectively managing algebraic expressions, especially those with exponents, can help solve equations more smoothly, making algebraic problem-solving more efficient and less error-prone.

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Most popular questions from this chapter

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\frac{3}{2}}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\frac{\sqrt[3]{m n} \cdot \sqrt[3]{m^{2}}}{\sqrt[3]{n^{2}}}$$

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{\sqrt{r}}{3-\sqrt{r}}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{343}$$

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{p}{\sqrt{p}+2}$$
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