Question

Simplify each expression. $$\left(\frac{27}{8}\right)^{2 / 3}$$

Short answer

The simplified expression is \(\frac{9}{4}\).

Step-by-step solution

Identify the Exponents and the Fraction

First, recognize that the expression involves a fraction raised to a power. Here, the fraction is \(\frac{27}{8}\) and the exponent is \( \frac{2}{3} \).

Apply the Power to Both Numerator and Denominator

Use the rule \(\bigg( \frac{a}{b} \bigg)^n = \frac{a^n}{b^n}\), where \(a = 27\), \(b = 8\), and \(n = \frac{2}{3}\). Thus, the expression can be rewritten as \(\frac{27^{2/3}}{8^{2/3}}\).

Simplify the Numerator

To simplify \(27^{2/3}\), recognize that \(27 = 3^3\). Therefore, \(27^{2/3} = (3^3)^{2/3} = 3^{3 \times \frac{2}{3}} = 3^2 = 9\).

Simplify the Denominator

To simplify \(8^{2/3}\), recognize that \(8 = 2^3\). Therefore, \(8^{2/3} = (2^3)^{2/3} = 2^{3 \times \frac{2}{3}} = 2^2 = 4\).

Simplify the Fraction

Now that you have simplified both the numerator and the denominator, the expression is \(\frac{9}{4}\). This fraction is already in its simplest form.

Key concepts

Exponent Rules

Understanding exponent rules is key to simplifying expressions like the one in our exercise. In mathematics, an exponent indicates how many times a number (the base) is multiplied by itself. For example, in the expression \(3^2\), the base is 3 and the exponent is 2, meaning 3 is multiplied by itself once: \(3 \times 3 = 9\).
Now, when dealing with expressions that involve fractions and exponents, certain rules become very handy.
Here are some critical rules:

• Product Rule: \(a^m \times a^n = a^{m+n}\). If the bases are the same, add the exponents.
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\). If the bases are the same, subtract the exponents.
• Power Rule: \((a^m)^n = a^{m \times n}\). When raising a power to another power, multiply the exponents.
• Distributive Property of Exponents: \(\big( \frac{a}{b} \big)^n = \frac{a^n}{b^n}\). Apply the exponent to both the numerator and the denominator.

Let's apply the Distributive Property in our exercise. The given expression is \(\big( \frac{27}{8} \big)^{2/3} \). According to the rule, it can be split into \(\frac{27^{2/3}}{8^{2/3}}\). This makes the expression easier to handle.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible. You aim to reduce them to a form where they can't be simplified any further. This often involves factoring, combining like terms, and applying exponent rules.
In the current exercise, we simplified the fraction \(\frac{27}{8}\) with a fractional exponent \(\frac{2}{3}\). We took these steps:

• First, recognize that both 27 and 8 are perfect cubes: 27 is \(3^3\) and 8 is \(2^3\).
• Next, apply the power rule to these bases. For the numerator: \(27^{2/3} = (3^3)^{2/3} = 3^{3 \times 2/3} = 3^2 = 9\). For the denominator: \(8^{2/3} = (2^3)^{2/3} = 2^{3 \times 2/3} = 2^2 = 4\).
• This results in \(\frac{9}{4}\), a simplified form of the original expression.

Every simplification step leverages exponent rules, showcasing the importance of mastering them. Always break your problem into smaller, more manageable parts and apply the learned rules to each one.

Fractional Powers

Fractional powers, also known as rational exponents, combine the concepts of roots and powers. An expression like \(a^{m/n}\) means taking the nth root of a raised to the m-th power: \(a^{m/n} = \big( \root(n)(a^m) \big)\).
For example, in \(27^{2/3}\), you first interpret 27 as \(3^3\). The fractional exponent \(\frac{2}{3}\) signifies taking the cube root of 27 and then squaring the result.
This double operation can be simplified step-by-step:

• Find the base: Recognize 27 as \(3^3\).
• Apply the fractional power: \((3^3)^{2/3}\).
• Use the power rule: \(3^{3 \times 2/3} = 3^2\).
• Finally, compute: \( 3^2 = 9\).

This method applies similarly to the denominator: interpret 8 as \(2^3\) and work through \(8^{2/3} = 4\).
Understanding fractional powers vs. whole-number exponents might take practice, but knowing they link directly into root and power rules will help you simplify expressions correctly.