Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$x^{3 / 4} x^{1 / 3} x^{-1 / 2}$$

Short answer

The simplified expression is \( x^{7/12} \).

Step-by-step solution

- Apply the Product of Powers Property

When multiplying expressions with the same base, add their exponents. Combine the exponents of all the terms: \( x^{3/4} x^{1/3} x^{-1/2} = x^{(3/4 + 1/3 + (-1/2))} \)

- Find a Common Denominator

To add the exponents, first find a common denominator. For 4, 3, and 2, the common denominator is 12. Convert each exponent: \( \frac{3}{4} = \frac{9}{12} \)\( \frac{1}{3} = \frac{4}{12} \)\( -\frac{1}{2} = -\frac{6}{12} \)

- Add the Exponents

Add the converted fractions: \( \frac{9}{12} + \frac{4}{12} - \frac{6}{12} \) Combine the numerators: \( 9 + 4 - 6 = 7 \). Hence, \( x^{(9/12 + 4/12 - 6/12)} = x^{7/12} \)

- Ensure Exponents are Positive

Since the final exponent \( \frac{7}{12} \) is already positive, no further changes are needed.

Key concepts

Product of Powers Property

When multiplying expressions with the same base, you can simplify them by adding their exponents. This is known as the 'Product of Powers Property'. For example, if you have the expression: \( x^{a} \cdot x^{b} \), you can combine them into a single power: \( x^{a + b} \). This property makes it easier to deal with complicated expressions by consolidating them into a simpler form. In the given exercise, we are combining the exponents of the terms \( x^{3/4} \), \( x^{1/3} \), and \( x^{-1/2} \) by adding them up.

Common Denominators in Fractions

To add or subtract fractions, they must have a common denominator. This means that the bottom number in each fraction should be the same. Finding a common denominator involves determining the Least Common Multiple (LCM) of the denominators. In our case, to add the exponents \( \frac{3}{4} \), \( \frac{1}{3} \), and \( -\frac{1}{2} \), we need a common denominator. The LCM of 4, 3, and 2 is 12. So, we convert each fraction:

• \( \frac{3}{4} = \frac{9}{12} \)
• \( \frac{1}{3} = \frac{4}{12} \)
• \( -\frac{1}{2} = -\frac{6}{12} \)

Now, we can easily add the fractions together.

Positive Exponents

Exponents can be positive or negative. Positive exponents indicate the number of times a number, called the base, is multiplied by itself. For example, \( x^3 \) means \( x \cdot x \cdot x \). Negative exponents, on the other hand, represent the reciprocal of the base raised to the opposite positive exponent. For example, \( x^{-2} \) means \( \frac{1}{x^2} \). In this exercise, we are required to express the final answer with only positive exponents. Since our result is \( x^{7/12} \), which is already positive, we don't need to change it.

Adding Fractions

Adding fractions requires a common denominator. Once fractions have the same denominator, you can add their numerators directly. For the exponents in this exercise, we added the converted fractions: \( \frac{9}{12} + \frac{4}{12} - \frac{6}{12} \). This simplifies to \( \frac{9 + 4 - 6}{12} = \frac{7}{12} \). The resulting fraction is \( \frac{7}{12} \), which is the exponent for the variable \( x \) in the simplified expression \( x^{7/12} \). This step is crucial for combining and simplifying terms in mathematical expressions.