Question

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

Short answer

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

Step-by-step solution

Understanding Negative Exponents

When dealing with negative exponents, we can use the rule \(\frac{1}{a^n} = a^{-n}\). This allows us to rewrite the expression with positive exponents. So, \(\big(\frac{8}{27}\big)^{-2/3}\) can be rewritten as \(\big(\frac{27}{8}\big)^{2/3}\).

Applying the Rational Exponent

The exponent \(\frac{2}{3}\) can be separated into two parts: the denominator (3), which indicates a root, and the numerator (2), which indicates a power. First, apply the cube root: \(\big(\frac{27}{8}\big)^{1/3}\).

Calculating the Cube Root

Find the cube root of both the numerator and the denominator separately. \(\big(\frac{27}{8}\big)^{1/3} = \frac{27^{1/3}}{8^{1/3}} = \frac{3}{2}\).

Applying the Square

Now, raise the result \(\frac{3}{2}\) to the power of 2. \(\big(\frac{3}{2}\big)^{2} = \frac{3^2}{2^2} = \frac{9}{4}\).

Key concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they are simply another way to express division. When we see a negative exponent, for instance, \(\frac{1}{a^n}\), it means the same as \(a^{-n}\). For example, \(5^{-3}\) is equivalent to \(\frac{1}{5^3}\). This rule helps us rewrite expressions with negative exponents into a more manageable form with positive exponents. In our exercise, \(\bigg(\frac{8}{27}\bigg)^{-2/3}\) was transformed into \(\bigg(\frac{27}{8}\bigg)^{2/3}\) by using this rule.
Breaking down expressions with negative exponents helps simplify them step by step and avoid confusion.

Rational Exponents

Rational exponents can be split into two components: the numerator and the denominator. The denominator of the rational exponent indicates the root, and the numerator indicates the power.
For instance, in \(\bigg(\frac{27}{8}\bigg)^{2/3}\), the \(2/3\) tells us to first take the cube root (because of the 3 in the denominator) and then raise the result to the power of 2 (because of the 2 in the numerator). This two-step approach makes it easier to handle complex expressions.

Cube Root

The cube root is the opposite of cubing a number; it is finding which number, when cubed, equals the initial number. For example, the cube root of 27 is 3, as \(3^3 = 27\).
In our problem, we had to find the cube root of both the numerator and the denominator. The cube root of 27 is 3, and the cube root of 8 is 2. By simplifying \(\bigg(\frac{27}{8}\bigg)^{1/3}\) to \(\frac{3}{2}\), we make the next steps much simpler to follow.

Fractional Exponents

Fractional exponents are a way to express both roots and powers in one compact form. For example, \(a^{m/n}\) means we take the n-th root of a and then raise it to the m-th power.
In our exercise, we dealt with \(\bigg(\frac{27}{8}\bigg)^{2/3}\) by first applying the cube root to get \(\frac{3}{2}\).
Then, we raised \(\frac{3}{2}\) to the power of 2 to get the final simplified form: \(\frac{9}{4}\). This demonstrates that separating the fractional exponent into root and power makes complex expressions much more manageable.