Question

Find: \((a) x^{3} / x^{-3}\) \((b)\left(x^{1 / 2} \times x^{1 / 3}\right) / x^{2 / 3}\)

Short answer

(a) \(x^6\); (b) \(x^{1/6}\).

Step-by-step solution

Simplify the Expression (a)

The expression to simplify is \( \frac{x^3}{x^{-3}} \). Use the property of exponents: \( a^{m}/a^{n} = a^{m-n} \). Subtract the exponent in the denominator from the exponent in the numerator: \( 3 - (-3) = 3 + 3 = 6 \). Therefore, the expression simplifies to \( x^6 \).

Simplify the Expression (b) - Multiplication

The expression is \( \left( x^{1/2} \times x^{1/3} \right) / x^{2/3} \). First, simplify the numerator using the property of exponents: \( a^{m} \times a^{n} = a^{m+n} \). So, \( x^{1/2} \times x^{1/3} = x^{(1/2 + 1/3)} \). Find a common denominator for the fractions: \( 1/2 + 1/3 = 3/6 + 2/6 = 5/6 \). The expression in the numerator becomes \( x^{5/6} \).

Simplify the Expression (b) - Division

Now, divide the simplified numerator by the denominator using the property of exponents: \( \frac{x^{m}}{x^{n}} = x^{m-n} \). Insert the exponents: \( \frac{x^{5/6}}{x^{2/3}} = x^{(5/6 - 2/3)} \). Convert \( 2/3 \) to have a common denominator as \( 5/6 \): \( 2/3 = 4/6 \). Thus, \( 5/6 - 4/6 = 1/6 \). The simplified expression is \( x^{1/6} \).

Key concepts

Properties of Exponents

Exponentiation is a powerful tool in mathematics used to simplify expressions or solve equations. Understanding the properties of exponents is crucial. These properties allow us to manage powers more effectively. Here are a few basic properties you should know:

• Product of Powers: This states that when you multiply two exponents with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: For dividing exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator: \(a^m / a^n = a^{m-n}\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).

In the original exercise, these properties are used to simplify expressions like \(x^3 / x^{-3}\). By applying the quotient of powers property, subtracting the negative exponent gives the result \(x^6\). This demonstrates how exponents simplify complex expressions easily.

Simplifying Expressions

Learning how to simplify expressions is an essential skill in algebra. It makes working with equations and functions much more manageable. When simplifying, we apply known rules or properties. For expressions like \((x^{1/2} \times x^{1/3}) / x^{2/3}\), we need to conduct operations systematically:

• Numerator Simplification: First, apply the product of powers property to combine terms: \(x^{1/2} \times x^{1/3} = x^{5/6}\). Here, finding a common denominator for the exponents \(1/2\) and \(1/3\) helps.


• Overall Simplification: Next, perform division using the quotient of powers property. Subtract the denominator exponent from the numerator: \(x^{5/6} / x^{2/3} = x^{1/6}\). Again, common denominators help with subtraction.

The step-by-step approach shows how simplifying works, combining multiple exponent rules to achieve easier forms.

Fractional Exponents

Fractional exponents appear daunting but are very intuitive once you understand their meaning. They represent roots and powers. This makes them incredibly useful for expressing roots in a compact form.

• What are Fractional Exponents? They are exponents that are fractions. The numerator indicates the power, while the denominator indicates the root.
• Interpreting Fractional Exponents: For instance, \(x^{1/2}\) is equivalent to the square root of \(x\), \(x^{1/3}\) is the cube root, and so on.

In understanding the original exercise, the expression \(x^{1/2} \times x^{1/3}\) simplifies to \(x^{5/6}\), using fractional exponent principles. This shows the combination of root and power operations. Simplifying further to \(x^{1/6}\) highlights how different exponents can fit together using addition and subtraction in fractional forms.In conclusion, mastering these concepts enhances mathematical clarity. It equips you with tools to tackle algebraic challenges more confidently.