Problem 28 - Algebra 2

Question

Evaluate each expression. $$ (-27)^{-\frac{2}{3}} $$

Step-by-Step Solution

Answer

The value of \((-27)^{-\frac{2}{3}}\) is \(\frac{1}{9}\).

Step 1: Understand Negative Exponent

The expression \( (-27)^{-\frac{2}{3}} \) includes a negative exponent, \(-\frac{2}{3}\). A negative exponent indicates a reciprocal. Hence, we can rewrite the expression as its reciprocal:\[ (-27)^{-\frac{2}{3}} = \frac{1}{(-27)^{\frac{2}{3}}} \]

Step 2: Root and Power Separation

The fractional exponent \(\frac{2}{3}\) indicates both a power and a root. Specifically, it's the cube root raised to the square. Let's separate these operations:\( (-27)^{\frac{2}{3}} = \left((-27)^{\frac{1}{3}}\right)^2 \)

Step 3: Calculate the Cube Root

Now, we need to find the cube root of \(-27\). Since the cube root of a negative number is negative, we have:\[ (-27)^{\frac{1}{3}} = -3 \]

Step 4: Square the Result

Next, take the result from the cube root calculation and square it:\[ (-3)^2 = 9 \]

Step 5: Apply Reciprocation

Finally, recall that due to the original negative exponent, we need to take the reciprocal of \(9\):\[ \frac{1}{9} \]

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a way to represent roots and powers in one concise expression. When you see an exponent expressed as a fraction, like \( \frac{a}{b} \), this indicates two operations:

• The denominator \( b \) refers to the root you need to take; specifically, the \( b \)-th root.
• The numerator \( a \) indicates that you should raise the result of the root to the \( a \)-th power.

For example, \( (-27)^{\frac{2}{3}} \) means you take the cube root of \(-27\) and then square the result. This dual operation can simplify solving many problems, allowing both powers and roots to be managed in a single step.

Cube Root

Calculating cube roots is essential when dealing with fractional exponents where the denominator is \(3\). The cube root of a number \( a \), expressed as \( a^{\frac{1}{3}} \), is the value which, when multiplied by itself three times, gives \( a \) back. Notably, the cube root of a negative number is also negative because the multiplication of three negative numbers results in a negative. For instance, the cube root of \(-27\) is \(-3\) because \(-3 \times -3 \times -3 = -27\). Knowing how to find cube roots is important when simplifying expressions with fractional exponents.

Reciprocal

A reciprocal of a number is simply \( \frac{1}{a} \) if the number is \( a \). In expressions with negative exponents, reciprocals play a crucial role. A negative exponent such as \(-\frac{2}{3}\) tells us to take the reciprocal of the base with the positive version of that exponent. For example, in the expression \((-27)^{-\frac{2}{3}}\), we find the reciprocal of \((-27)^{\frac{2}{3}}\) after solving for the positive exponent, which effectively inverts the result. Reciprocals help us transform complex exponent rules into manageable calculations.

Exponentiation Rules

Understanding exponentiation rules is essential for correctly solving expressions involving powers. Key rules include:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \times n} \)
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)
• Fractional Exponents Rule: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)

Using these rules, we can systematically approach a problem, like breaking down \((-27)^{-\frac{2}{3}}\) into simpler steps by converting the negative exponent and applying the fractional exponent rule to separate power and root operations.