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Chapter 0: Problem 17

Simplify the expression. \(\sqrt{\frac{4}{49}}\)

Short Answer

Expert verified
The simplified form is \( \frac{2}{7} \).

Step by step solution

01

Identify the Square Roots

Recognize that you need to find the square root of both the numerator and the denominator separately. The expression inside the square root is \(\frac{4}{49}\).
02

Find the Square Root of the Numerator

Calculate the square root of the numerator. \(\text{The numerator is 4, and } \sqrt{4} = 2\).
03

Find the Square Root of the Denominator

Next, calculate the square root of the denominator. \(\text{The denominator is 49, and } \sqrt{49} = 7\).
04

Simplify the Expression

Combine the square roots of the numerator and the denominator. \(\frac{\text{Square root of the numerator}}{\text{Square root of the denominator}} = \frac{2}{7}\).

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Key Concepts

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

Simplifying Expressions
Simplifying mathematical expressions is an essential skill in algebra. When we simplify an expression, we make it easier to understand or use. One common example involves working with square roots and fractions.

For instance, let鈥檚 look at the expression \( \sqrt{\frac{4}{49}} \). To simplify this, we follow a straightforward approach:
  • Identify components: numerator and denominator.
  • Determine the square roots of both parts separately.
  • Combine the simplified parts.
Let鈥檚 break it down further to deepen understanding!
Understanding Radicals
Radicals, often seen as 鈭� in mathematics, represent square roots, cube roots, and higher-order roots. When simplifying radicals like \( \sqrt{\frac{4}{49}} \), you need to recognize each part under the radical.

Here, we have two parts under the square root: the numerator (4) and the denominator (49). Both parts need to be dealt with separately, because \( \sqrt{\frac{a}{b}} \ = \frac{\sqrt{a}}{\sqrt{b}} \).

You calculate the square root of 4, which is 2, and the square root of 49, which is 7. Then, you combine these two results to get the simplified form, \( \frac{2}{7} \).
Numerator and Denominator
In fractions, the **numerator** is the top number, representing how many parts we have.

The **denominator** is the bottom number, showing the total number of equal parts. For \( \frac{4}{49} \), 4 is the numerator, and 49 is the denominator.

Understanding these terms is crucial when dealing with fractions inside square roots. When you simplify \( \sqrt{\frac{4}{49}} \, \ \frac{2}{7} \), it means finding the square root of each part separately and then combining them back into a simplified fraction.

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Most popular questions from this chapter

Simplify each expression. $$ (y+7)^{2}-2(y-3)^{2} $$

If \(n<0,\) then \(-|n|=\) _____.

Determine the sign of the expression. Assume that \(a, b,\) and \(c\) are real numbers and \(a<0, b>0,\) and \(c<0\). $$\frac{a^{2} c}{b^{4}}$$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 5 \sqrt{2}(2 \sqrt{2}+6 \sqrt{3}-4) $$

Explain the similarity in simplifying the given expressions. a. \((3 x+2)(4 x-7)\) b. \((3 \sqrt{x}+\sqrt{2})(4 \sqrt{x}-\sqrt{7})\)
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