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Chapter 8: Problem 25

Simplify the squares and square roots. $$7^{2}$$

Short Answer

Expert verified
49

Step by step solution

01

Understand Exponents

An exponent indicates that a number (the base) is to be multiplied by itself a certain number of times. Here, the base is 7, and the exponent is 2.
02

Multiply the Base by Itself

To simplify the expression, multiply the base number by itself according to the exponent: \[7^2 = 7 \times 7\].
03

Calculate the Result

Perform the multiplication: \[7 \times 7 = 49\].
04

Write the Final Answer

The simplified form of the expression is \[7^2 = 49\].

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in mathematics that help us to simplify repeated multiplication. An exponent, written as a small number to the top right of a base number, tells us how many times to multiply the base number by itself. For instance, in the expression \(7^2\), 7 is the base and 2 is the exponent. This means we multiply 7 by itself 2 times. Exponents are especially useful for expressing large numbers compactly.
Here鈥檚 a breakdown of how exponents work:
  • \(7^2\) means 7 multiplied by itself 2 times.
  • \(7 \times 7 = 49\)
Understanding this concept can simplify many mathematical calculations, making them more manageable and easier to perform.
Multiplication in Exponents
When working with exponents, multiplication plays a key role. The exponent tells you how many times to use the base as a factor in a multiplication. In our example \(7^2\), we take the base (7) and multiply it by itself according to the exponent (2), i.e., \(7 \times 7\).
Here is a step-by-step explanation:
  • First, identify the base and the exponent: In \(7^2\), 7 is the base and 2 is the exponent.
  • Then, multiply the base by itself as many times as indicated by the exponent: \(7 \times 7 = 49\).
This process transforms exponential notation into a simple multiplication problem, making it easier to understand and solve.
Base and Exponent Relationship
The base and exponent collaborate to express repeated multiplication succinctly. In any exponential expression, the base is the number being multiplied, and the exponent indicates how many times the base is used as a factor. For example, in \(7^2\), the base is 7, and the exponent is 2.
Here鈥檚 how to understand their relationship better:
  • The base (7) is the fundamental value that will be repeatedly multiplied.
  • The exponent (2) specifies how many times to multiply the base by itself.
  • This results in the calculation: \(7^2 = 7 \times 7 = 49\).
Recognizing this relationship between the base and exponent simplifies comprehension of how numbers are operated on in exponential form. It also highlights the efficiency of exponents in mathematical notation.

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Most popular questions from this chapter

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