Simplify (5x^(1/2))/(4x^(2/3))

Solution:

Step 1:

Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ to rewrite $x^{\frac{1}{2}}$ in the denominator: $\frac{5}{4x^{\frac{2}{3}}x^{-\frac{1}{2}}}$.

Step 2:

Combine the exponents by adding them together.

Step 2.1:

Reposition $x^{-\frac{1}{2}}$: $\frac{5}{4(x^{\frac{2}{3}}x^{-\frac{1}{2}})}$.

Step 2.2:

Use the exponent addition rule $a^{m}a^{n} = a^{m+n}$: $\frac{5}{4x^{\frac{2}{3} - \frac{1}{2}}}$.

Step 2.3:

Convert $-\frac{1}{2}$ to a fraction with a denominator of 6 by multiplying by $\frac{3}{3}$: $\frac{5}{4x^{\frac{-3}{6} + \frac{2}{3}}}$.

Step 2.4:

Convert $\frac{2}{3}$ to a fraction with a denominator of 6 by multiplying by $\frac{2}{2}$: $\frac{5}{4x^{\frac{-3}{6} + \frac{4}{6}}}$.

Step 2.5:

Combine the fractions over a common denominator of 6.

Step 2.6:

Add the numerators: $\frac{5}{4x^{\frac{-3 + 4}{6}}}$.

Step 2.7:

Simplify the exponent by calculating the numerator: $\frac{5}{4x^{\frac{1}{6}}}$.

Knowledge Notes:

To simplify an expression involving exponents, we can use several rules of exponents:

1. Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$, which allows us to move factors from the numerator to the denominator or vice versa by changing the sign of the exponent.

2. Power Rule: $a^{m}a^{n} = a^{m+n}$, which states that when multiplying like bases, we add the exponents.

3. Common Denominator: When adding or subtracting fractions with different denominators, we find a common denominator to combine the fractions.

In this problem, we also use the concept of equivalent fractions to rewrite exponents with a common denominator, which allows us to add or subtract them more easily. This is particularly useful when dealing with fractional exponents.