Simplify (4 square root of 4)/(3 square root of 5)

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: $1.19256958\dots$

Knowledge Notes:

To simplify the expression $\frac{4\sqrt{4}}{3\sqrt{5}}$, we follow these steps:

1. Radical Simplification: We simplify the square root of perfect squares. $\sqrt{4}$ simplifies to 2 because $2^2=4$.

2. 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.

3. 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 $(a^m{)}^n=a^{mn}$.

4. Simplifying Exponents: When we have the same base with an exponent of 1, we can simplify it to just the base, as $a^1=a$.

5. 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.