Evaluate (4x)/(9x^(1/2))

Solution:

Step 1:

Apply the rule for negative exponents, which states $a^{-n} = \frac{1}{a^{n}}$, to rewrite $x^{\frac{1}{2}}$ in the denominator as $x^{-\frac{1}{2}}$ in the numerator.

$$\frac{4x \cdot x^{-\frac{1}{2}}}{9}$$

Step 2:

Combine the terms with like bases by adding their exponents.

Step 2.1:

Position $x^{-\frac{1}{2}}$ next to $x$.

$$\frac{4(x^{-\frac{1}{2}} \cdot x)}{9}$$

Step 2.2:

Perform the multiplication of $x^{-\frac{1}{2}}$ and $x$.

Step 2.2.1:

Express $x$ with an exponent of $1$.

$$\frac{4(x^{-\frac{1}{2}} \cdot x^{1})}{9}$$

Step 2.2.2:

Utilize the exponent rule $a^{m} \cdot a^{n} = a^{m+n}$ to combine the exponents.

$$\frac{4x^{-\frac{1}{2} + 1}}{9}$$

Step 2.3:

Express the number $1$ as a fraction with a denominator of $2$ to match the denominator of the other exponent.

$$\frac{4x^{-\frac{1}{2} + \frac{2}{2}}}{9}$$

Step 2.4:

Sum the numerators while keeping the common denominator.

$$\frac{4x^{\frac{-1 + 2}{2}}}{9}$$

Step 2.5:

Add the numbers $-1$ and $2$ in the exponent.

$$\frac{4x^{\frac{1}{2}}}{9}$$

Knowledge Notes:

To solve the given expression $\frac{4x}{9x^{\frac{1}{2}}}$, we employ several algebraic rules and properties:

1. Negative Exponent Rule: This rule states that $a^{-n} = \frac{1}{a^{n}}$. It allows us to move a factor from the denominator to the numerator by changing the sign of its exponent.

2. Multiplication of Like Bases: When multiplying expressions with the same base, we add their exponents, according to the rule $a^{m} \cdot a^{n} = a^{m+n}$.

3. Fractional Exponents: A fractional exponent, such as $x^{\frac{1}{2}}$, represents a root of the base number. Specifically, $x^{\frac{1}{2}}$ is equivalent to $\sqrt{x}$.

4. Combining Fractions: When combining fractions with different numerators but the same denominator, we simply add or subtract the numerators while keeping the common denominator.

By applying these rules, we can simplify the given expression to a more manageable form without altering its value.