Problem 24 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-27)^{2 / 3} $$

Step-by-Step Solution

Answer

The value of \[ (-27)^{2/3} \] is 9.

Step 1: Understand the Problem

The expression \[ (-27)^{2/3} \]involves a negative number raised to a fractional power. The fractional exponent \( \frac{2}{3} \) indicates taking the cube root followed by squaring.

Step 2: Simplify the Fractional Exponent

Rewrite the expression \[ (-27)^{2/3} \]as \[ \left( (-27)^{1/3} \right)^2 \].This represents the cube root of \(-27\) raised to the 2nd power.

Step 3: Calculate the Cube Root

Find the cube root of \(-27\), which is \[ (-27)^{1/3} = -3 \],because \[ (-3)^3 = -27 \].

Step 4: Square the Result

Take the result from the cube root and square it:\[ (-3)^2 = 9 \].

Key Concepts

Cube Roots

Cube roots are a way to reverse cube operations. Thinking about it, a cube root helps us find a number which, when multiplied by itself three times, gives the original number. For instance, the cube root of \(-27\) is \-3\. This is because \((-3) \times (-3) \times (-3) = -27\). Cube roots are especially interesting because they can involve negative numbers, as cube operations involving negative numbers still produce negative results.

When dealing with cube roots, here are some things to remember:

• If a number is negative, like \(-27\), its cube root will also be negative.
• The symbol for cube root is often represented with an exponent of \(1/3\)
• Cube roots are one of the simplest roots because they work with both positive and negative numbers easily.

Negative Numbers in Exponents

Exponents with negative numbers can sometimes be surprising. When we talk about exponents, we're often thinking about multiplying a number by itself. But what happens when that number is negative? Let's think of \(-27\). If we raise this number to a power, like in the case \((-27)^{2/3}\), the negative base here is well-managed because the exponent \(2/3\) tells us to perform operations, like taking a cube root and then squaring, separately.

Here are some helpful pointers:

• If the exponent involves a fraction like \(2/3\), solve in parts – take the root first, then raise to the remaining power.
• Don't forget that if a number is negative, its cube operations generally still result in negative numbers.
• Be mindful that other operations can change negativity depending on how the power parts are handled, especially squaring.

Exponentiation Rules

Exponentiation rules help make complex problems easier by applying specific operations. When dealing with fractional exponents like \(\frac{2}{3}\), it's really about understanding the separate parts involved: the root and the power. Thinking of the expression \((-27)^{2/3}\) invites these key exponent rules:

• First, break down the exponent: \(a^{m/n}\) means take the n-th root of \(a\) and then raise the result to the m-th power.
• In this case, calculate the cube root first, turning \((-27)^{1/3}\) to \(-3\). Afterwards, you square the result: \((-3)^2\) equals 9.
• Remember, squaring a number makes it positive, hence the final result becomes positive even if the base was initially negative.

By fluently applying these rules, we grow to manage challenging problems with ease, focusing on operations one step at a time.