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Chapter 1: Problem 59

Perform the indicated operations. If \(a^{3}=5,\) then what does \(a^{12}\) equal?

Short Answer

Expert verified
The value of \(a^{12}\) is 625.

Step by step solution

01

Understanding the Problem

We need to find the value of \(a^{12}\) given that \(a^3 = 5\). This requires us to find a connection between \(a^3\) and \(a^{12}\).
02

Expressing Powers

We know that \(a^{12}\) can be expressed as a power of \(a^3\). Specifically, \(a^{12} = (a^3)^4\) because when you multiply the exponent 3 by 4, you get 12.
03

Substitution and Simplification

Since \(a^3 = 5\), substitute 5 into the expression for \(a^{12}\). Thus, \(a^{12} = (a^3)^4 = 5^4\).
04

Calculating the Power of 5

To find \(5^4\), compute it step-by-step: - Start with 5 multiplied by itself: - \(5 \times 5 = 25\) - \(25 \times 5 = 125\) - \(125 \times 5 = 625\)Thus, \(5^4 = 625\).
05

Concluding the Solution

We have computed that \(a^{12} = 625\), which gives us the solution based on the initial condition \(a^3 = 5\).

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Key Concepts

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

Exponentiation Techniques
Understanding exponentiation is crucial because it simplifies complex calculations. **Exponentiation** involves raising a base number to a power or exponent, which tells us how many times to multiply the base by itself. When working with exponents, there are several techniques to consider:
  • **Multiplying Powers:** When you multiply powers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
  • **Raising a Power to Another Power:** When raising an exponent to another exponent, multiply the exponents together. In our problem, \(a^{12} = (a^3)^4\), this means multiply 3 by 4 to get 12.
  • **Power of a Product:** To raise a product to a power, apply the exponent to each factor. That is \((ab)^n = a^n \times b^n\).
Mastering these techniques allows you to work with large numbers efficiently and tackle complex algebraic expressions with ease.
Mathematical Operations
Mathematical operations play a significant role in solving exponentiation problems. Operations such as multiplication and power calculation can greatly affect the outcome. Here's a closer look at their application in our problem:
  • **Multiplication:** This straightforward operation is used repeatedly in calculating powers. For example, to compute \(5^4\), repeatedly multiply 5 by itself.
  • **Exponentiation Calculation:** Directly involves multiplying a number by itself multiple times as in \(5^4 = 5 \times 5 \times 5 \times 5\).
Being aware of these operations helps simplify equations and solve them quickly. Attention to detailed computation, such as ensuring each multiplication step is correctly executed, ensures precision in the results.
Algebraic Manipulations
Algebraic manipulation skills are essential for restructuring and simplifying expressions. They are particularly useful in problems involving exponents, like converting \(a^{12}\) in terms of \(a^3\). Here鈥檚 how algebraic manipulations help:
  • **Rewriting Expressions:** Transform expressions to make calculations easier. Knowing that \(a^{12}=(a^3)^4\) helps transition to a simpler form.
  • **Substitution:** Replacing expressions with known values simplifies solving. Substituting \(a^3 = 5\) provides a clear path to finding \(a^{12}\).
  • **Simplification:** Break down expressions into manageable calculations. Compute \(5^4\) step-by-step for accuracy.
By mastering these manipulations, you gain the ability to approach mathematical problems systematically, leading to clearer and logical solutions.

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Most popular questions from this chapter

In using aircraft radar, the expression \((2 R-X)^{2}-\left(R^{2}+X^{2}\right)\) arises. Simplify this expression.

Set up an appropriate equation and solve. Data are accurate to two sig. digits unless greater accuracy is given. A car's radiator contains 12 L of antifreeze at a \(25 \%\) concentration. How many liters must be drained and then replaced by pure antifreeze to bring the concentration to \(50 \%\) (the manufacturer's "safe" level)?

Solve the given equations. $$1.5 x-0.3(x-4)=6$$

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