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Chapter 1: Problem 43

Find the following roots. See Examples 7 and \(8 .\) $$ \sqrt{49} $$

Short Answer

Expert verified
The square root of 49 is 7.

Step by step solution

01

Understanding the Problem

We need to find the square root of the number 49. The square root of a number is a value that, when multiplied by itself, gives the original number.
02

Identify the Perfect Square

49 is a perfect square because there is an integer that, when multiplied by itself, equals 49.
03

Determine the Square Root

Identify which integer squared results in 49. We know that \(7 \times 7 = 49\). Thus, the square root of 49 is 7.
04

Verify the Solution

Verify by squaring the result: \(7^2 = 49\), confirming that the square root of 49 is indeed 7.

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Key Concepts

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

Perfect Squares
A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For example, numbers like 1, 4, 9, 16, 25, 36, 49, and so forth are all perfect squares. This is because:
  • 1 is the result of \(1 \times 1\).
  • 4 is the result of \(2 \times 2\).
  • 9 is the result of \(3 \times 3\).
  • 16 is the result of \(4 \times 4\).
49 is a perfect square because the integer 7 multiplied by itself gives 49. Recognizing perfect squares is critical in simplifying square roots and other algebraic expressions. Knowing which numbers are perfect squares can help you quickly find roots without calculator assistance.
Integer Multiplication
Integer multiplication involves taking two whole numbers and calculating their product. In our problem, integer multiplication helps us find the root of a perfect square. When we say that 49 is a perfect square, we mean it is the product of the integer 7 multiplied by itself, i.e., \(7 \times 7 = 49\).

This is a straightforward process:
  • Select an integer 鈥� for 49, it's 7.
  • Multiply this integer by itself to see if you get the original number.
In this case, knowing that the multiplication \(7 \times 7\) results in 49 validates our identification of the integer required to find the square root of 49.
Mathematical Verification
Verification in mathematics is an essential part of problem-solving. It involves re-checking your result to ensure it is accurate. In our exercise with the square root of 49, verifying means taking the calculated square root, 7, and squaring it to see if it returns to the original number, 49.

Here's how you verify:
  • Take the calculated square root: 7.
  • Square it (multiply it by itself): \(7^2 = 7 \times 7 = 49\).
Through verification, we double-check that our answer is correct. This process not only builds confidence but also strengthens understanding of the mathematical relationship between numbers and their roots.

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Most popular questions from this chapter

Find the value of the expression when \(x_{1}=2, x_{2}=4, y_{1}=-3\) \(y_{2}=2\) $$ \frac{y_{2}-y_{1}}{x_{2}-x_{1}} $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ 1.5 x+2.3-0.7 x-5.9 $$

Use a calculator to approximate each square root. simplify the expression. Round answers to four decimal places. \(\sqrt{7.9}\)

Translating Write each phrase as an algebraic expression. Use the variable \(x\) to represent each unknown number. See Example 8 . Four, subtracted from three times a number.

Explain why \(-3^{2}\) and \((-3)^{2}\) simplify to different numbers.
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