Simplify (4 square root of 4)/(3 square root of 5)
The problem is asking you to simplify a given rational expression which involves square roots. Specifically, you are to simplify the fraction where the numerator is four times the square root of four, and the denominator is three times the square root of five. Simplification in this context typically involves reducing the expression to its simplest form, which may include rationalizing the denominator if necessary and simplifying any square roots that can be reduced to whole numbers.
$\frac{4 \sqrt{4}}{3 \sqrt{5}}$
Answer
Solution:
Step 1: Simplify the numerator.
- Step 1.1: Express 4 as $2^2$. $\frac{4 \sqrt{2^2}}{3 \sqrt{5}}$
- Step 1.2: Extract terms from under the radical, assuming all are positive real numbers. $\frac{4 \cdot 2}{3 \sqrt{5}}$
Step 2: Calculate the product of 4 and 2. $\frac{8}{3 \sqrt{5}}$
Step 3: Rationalize the denominator by multiplying by $\frac{\sqrt{5}}{\sqrt{5}}$. $\frac{8}{3 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$
Step 4: Simplify the denominator.
Step 4.1: Multiply the numerators and denominators. $\frac{8 \sqrt{5}}{3 \sqrt{5} \sqrt{5}}$
Step 4.2: Group the square roots. $\frac{8 \sqrt{5}}{3 (\sqrt{5} \sqrt{5})}$
Step 4.3: Represent $\sqrt{5}$ as a power. $\frac{8 \sqrt{5}}{3 ((\sqrt{5})^1 \sqrt{5})}$
Step 4.4: Repeat the representation for the other $\sqrt{5}$. $\frac{8 \sqrt{5}}{3 ((\sqrt{5})^1 (\sqrt{5})^1)}$
Step 4.5: Apply the power rule for multiplication $a^{m} a^{n} = a^{m + n}$. $\frac{8 \sqrt{5}}{3 (\sqrt{5})^{1 + 1}}$
Step 4.6: Add the exponents. $\frac{8 \sqrt{5}}{3 (\sqrt{5})^2}$
Step 4.7: Convert $(\sqrt{5})^2$ to 5.
Step 4.7.1: Use the radical to power conversion $\sqrt[n]{a^x} = a^{\frac{x}{n}}$. $\frac{8 \sqrt{5}}{3 ((5^{\frac{1}{2}})^2)}$
Step 4.7.2: Apply the power rule for exponents $(a^m)^n = a^{mn}$. $\frac{8 \sqrt{5}}{3 \cdot 5^{\frac{1}{2} \cdot 2}}$
Step 4.7.3: Simplify the exponents. $\frac{8 \sqrt{5}}{3 \cdot 5^{\frac{2}{2}}}$
Step 4.7.4: Simplify the fraction.
- Step 4.7.4.1: Simplify the exponent. $\frac{8 \sqrt{5}}{3 \cdot 5^{\cancel{2}/\cancel{2}}}$
- Step 4.7.4.2: Rewrite the expression. $\frac{8 \sqrt{5}}{3 \cdot 5^1}$
Step 4.7.5: Evaluate the exponent. $\frac{8 \sqrt{5}}{3 \cdot 5}$
Step 5: Multiply 3 by 5. $\frac{8 \sqrt{5}}{15}$
Step 6: Present the result in various forms.
- Exact Form: $\frac{8 \sqrt{5}}{15}$
- Decimal Form: $1.19256958 \ldots$
Knowledge Notes:
To simplify the expression $\frac{4 \sqrt{4}}{3 \sqrt{5}}$, we follow these steps:
Radical Simplification: We simplify the square root of perfect squares. $\sqrt{4}$ simplifies to 2 because $2^2 = 4$.
Rationalizing the Denominator: To eliminate the radical from the denominator, we multiply the fraction by a form of 1 that contains the radical in both the numerator and the denominator.
Power Rules: We use the power rule for radicals and exponents to simplify expressions. The power rule states that $a^{m} a^{n} = a^{m + n}$ and $(a^m)^n = a^{mn}$.
Simplifying Exponents: When we have the same base with an exponent of 1, we can simplify it to just the base, as $a^1 = a$.
Multiplication and Division: We perform multiplication and division as normal, simplifying the fraction to its lowest terms if possible.
By following these steps, we can simplify radical expressions and rationalize denominators, which is a common requirement in algebra to present answers in their simplest form.
