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Chapter 6: Problem 4

Evaluate the expressions without using a calculator. $$ 5^{1} \cdot 1^{4} \cdot 3^{2} $$

Short Answer

Expert verified
Question: Evaluate the expression $5^1 \cdot 1^4 \cdot 3^2$ without using a calculator. Answer: 45

Step by step solution

01

Evaluate the exponential expressions

First, we will evaluate each exponential expression: $$ 5^1 = 5 \\ 1^4 = 1 \\ 3^2 = 9 $$
02

Perform the multiplication

Now that we have the results of the exponential expressions, we can proceed to multiply them together: $$ 5^1 \cdot 1^4 \cdot 3^2 = 5 \cdot 1 \cdot 9 $$
03

Simplify the multiplication

Our final step is to simplify the multiplication: $$ 5 \cdot 1 \cdot 9 = 5 \cdot 9 = 45 $$ So, the expression evaluates to 45 without using a calculator.

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Key Concepts

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

Evaluate Expressions
Evaluating expressions is the process of finding the value of mathematical statements. In this exercise, we have an expression made up of exponential components that need to be evaluated separately before multiplying them together. Let's break it down:
  • Exponential expressions involve numbers raised to a power, expressed as \( a^b \), where \( a \) is the base and \( b \) is the exponent.
  • The first expression, \( 5^1 \), asks us to raise 5 to the first power. This is straightforward, as any number to the power of one is itself, so \( 5^1 = 5 \).
  • The second expression, \( 1^4 \), involves 1 raised to the fourth power. Because 1 raised to any power remains 1, \( 1^4 = 1 \).
  • The third is \( 3^2 \), which means 3 is raised to the power of 2. So, \( 3 \times 3 = 9 \), resulting in \( 3^2 = 9 \).
By evaluating each exponential expression, we can simplify the process and get ready for the multiplication step.
Multiplication in Algebra
Multiplication in algebra follows straightforward principles, similar to numerical multiplication. Once we've evaluated each part of the problem, we can multiply these together to find our final answer. Start by reviewing your results from the exponential step:
  • We have \( 5 \), which came from \( 5^1 \).
  • Then, there's \( 1 \), derived from \( 1^4 \).
  • Finally, \( 9 \) from \( 3^2 \).
These components are then multiplied in sequence: \( 5 \times 1 \times 9 \).
Multiplication in algebra respects the commutative property, which means the order doesn't matter. Thus, you can solve it as \( 5 \times 9 = 45 \), and since \( 5 \times 1 \) is just \( 5 \), it simplifies the process to form the final product of 45.
Simplifying Expressions
Simplifying expressions is all about combining terms and operations until we achieve the simplest form or the final answer. In algebra, we often aim to make expressions as uncomplicated as possible for easier problem solving. To simplify the given problem, follow these steps:
  • First, always evaluate exponentials as you did previously: \( 5^1 = 5 \), \( 1^4 = 1 \), and \( 3^2 = 9 \).
  • Then, carry out the multiplication: \( 5 \cdot 1 \cdot 9 \).
  • Since multiplying by 1 does not change the value, simplify even further to \( 5 \times 9 \).
  • Calculate \( 5 \times 9 \) to get your final answer, 45.
Remember, simplifying expressions can make them much easier to work with, especially in complex problems, by focusing on significant operations and reducing unnecessary components.

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Most popular questions from this chapter

Combine radicals, if possible. \(\sqrt[3]{4 x}+6 \sqrt[3]{4 x}-2 \sqrt[3]{4 x}\)

Combine radicals, if possible. \(2 \sqrt{3}+\frac{\sqrt{3}}{2}\)

Combine radicals, if possible. \(5 \sqrt{9}-2 \sqrt{144}\)

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{6 g^{5}}{7 h^{7}}\right)^{2} $$

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