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Chapter 9: Problem 67

Simplify all radicals, and combine like terms. Express fractions in lowest terms. \(\frac{4}{5}+\sqrt{\frac{48}{25}}\)

Short Answer

Expert verified
ewline \[ \frac{4(1 + \sqrt{3})}{5} \]

Step by step solution

01

- Simplify the radical

Express the term inside the square root in simpler form. ewline ewline \text{ Simplify } \frac{48}{25} \text{ since both 48 and 25 are perfect squares.}ewline \[ \frac{48}{25} = \frac{16 \times 3}{25} = \frac{16}{25} \times 3 \]ewline \[ \text{ We can apply the property of square roots: } \sqrt{\frac{a \times b}{c}} = \frac{\sqrt{a} \times \sqrt{b}}{\sqrt{c}} \]ewline \[ \sqrt{\frac{48}{25}} = \frac{\sqrt{16 \times 3}}{5} = \frac{\sqrt{16} \times \sqrt{3}}{5} = \frac{4\sqrt{3}}{5} \]
02

- Combine like terms

Now, add the simplified radical to the fraction. ewline \[ \frac{4}{5}+\frac{4\sqrt{3}}{5} = \frac{4 + 4\sqrt{3}}{5} \]ewline \[ \frac{4 + 4\sqrt{3}}{5} = \frac{4(1 + \sqrt{3})}{5} \]

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Key Concepts

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

Combining Like Terms
Combining like terms is essential for simplifying expressions. Like terms are terms that have the same variable raised to the same power. In our problem, we start with \(\frac{4}{5} + 袄蹿谤补肠调4袄迟别虫迟调鈭殅3皑调5皑\).

Here, both terms have a common denominator of 5, making them like terms. To combine them, simply add the numerators while keeping the denominator the same:
  • \frac{4}{5} + 袄蹿谤补肠调4袄迟别虫迟调鈭殅3皑调5皑 = \frac{4+4\text{鈭殅3}{5}
You can notice that both terms have a common factor of 4 in the numerator. Factoring 4 out, we get:
  • 袄蹿谤补肠调4(1+袄迟别虫迟调鈭殅3)皑调5皑
This expression is now simplified, showing that combining like terms helps to make expressions concise and easy to interpret.
Properties of Square Roots
Understanding the properties of square roots helps simplify complex expressions. One key property is:
  • \text{\text{鈭殅\frac{a}{b}} = \frac{\text{鈭殅a}{\text{鈭殅b}
In our exercise, we simplify \(\text{鈭殅\frac{48}{25}\). Notice that 48 can be rewritten as 16 脳 3, and 25 is already a perfect square:
  • \frac{48}{25} = \frac{16 \times 3}{25} = \frac{16}{25} \times 3
  • \text{鈭殅\frac{48}{25} = \frac{\text{鈭殅(16 \times 3)}{5} = \frac{\text{鈭殅16 \times \text{鈭殅3}{5}
Since \(\text{鈭殅16 = 4\), we further simplify to:

袄蹿谤补肠调4袄迟别虫迟调鈭殅3皑调5皑

Thus, applying the property of square roots can make radical expressions easier to handle and incorporate into larger equations.
Expressing Fractions in Simplest Form
Expressing fractions in their simplest form makes them clearer and more manageable. The simplest form of a fraction is where the numerator and the denominator have no common factors other than 1.

In our exercise, the final expression is already presented in its simplest form after combining like terms: \(袄蹿谤补肠调4(1+袄迟别虫迟调鈭殅3)皑调5皑\).

Here are some simple steps to ensure fractions are in their simplest form:
  • Factor both the numerator and the denominator completely.
  • Divide both by their Greatest Common Divisor (GCD), if any.
For example:

\frac{24}{36} would be simplified by finding the GCD, which is 12, and dividing both terms by 12:
  • \frac{24 \div 12}{36 \div 12} = \frac{2}{3}
Applying these principles ensures clarity and precision in mathematical expressions.

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Most popular questions from this chapter

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 2 x^{2}+x+3=0 $$

Solve each equation by completing the square. Give (a) exact solutions and (b) solutions rounded to the nearest thousandth. $$(x+1)(x+3)=2 $$

Solve equation by using the square root property. Simplify all radicals. \((m+2)^{2}=-17\)

Solve each problem. Solve the formula \(S=2 \pi r h+\pi r^{2}\) for \(r\) by writing it in the form \(a r^{2}+b r+c=0\) and then using the quadratic formula. (Leave \(\pm\) in your answer.

Given \(f(x)=2 x^{2}-5 x+3,\) find each of the following. $$ f(-2) $$
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