Problem 47 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(-\frac{1}{27}\right)^{-5 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \( -243 \).

Step 1: Understanding Negative Exponents

The expression is \( \left(-\frac{1}{27}\right)^{-5 / 3} \). A negative exponent indicates that the base should be reciprocated, or inverted, and then raised to the positive exponent. So, \( a^{-b} = \frac{1}{a^b} \).

Step 2: Convert the Negative Exponent

First, reciprocate the base: \( \left(-\frac{1}{27}\right)^{-5/3} = \left(-27 \right)^{5/3} \). This inverts the base from \( -\frac{1}{27} \) to \( -27 \).

Step 3: Simplify the Base with the Fractional Exponent (Cube Root Step)

The exponent \( 5/3 \) means we should first take the cube root and then raise it to the power of 5. The cube root of \( -27 \) is \( -3 \), as \( (-3)^3 = -27 \).

Step 4: Apply the Power of the Exponent

Now raise \( -3 \) to the power of 5. Calculate: \[ (-3)^5 = (-3) \times (-3) \times (-3) \times (-3) \times (-3) = -243. \]

Step 5: Final Result

Thus, \( \left(-\frac{1}{27}\right)^{-5 / 3} \) simplifies to \( -243 \).

Key Concepts

Fractional Exponents

Fractional exponents can seem complex, but they basically break down operations into simpler steps. A fractional exponent like \( a^{m/n} \) means you perform two operations: taking the \( n \)-th root and then raising the result to the \( m \)-th power. The numerator \( m \) is the power and the denominator \( n \) is the root. This is essential to simplify expressions without a calculator.
For example, in \( \, (-27)^{5/3} \, \): you first take the cube root of \( -27 \), resulting in \( -3 \), and then take that result, \( -3 \), and raise it to the 5th power.
Remember:

• The power (numerator) tells you how many times to multiply the number by itself after taking the root.
• The root (denominator) tells you which type of root (e.g., square root, cube root) to take first.
• This two-step process breaks down complex calculations into more manageable parts.

Reciprocal of a Base

Understanding the reciprocal of a base when dealing with negative exponents is crucial. A negative exponent flips the base number. Specifically, \( a^{-b} = \frac{1}{a^b} \). This means you take the reciprocal, or inverse, of the base and then apply the new positive exponent.
In \( \, \left(-\frac{1}{27}\right)^{-5 / 3} \, \), first step involves reciprocating the base. Hence, \( \left(-\frac{1}{27}\right) \) becomes \( \left(-27\right) \). This transformation flips the fraction and changes the base from part of a fraction to its whole number counterpart. Always remember that if your original base is a fraction, flipping it due to a negative exponent gives you the inverted base.
This operation of inverting the base helps handle negative exponents more easily, setting up the problem for solving the power calculation accurately.

Cube Root

Taking the cube root is a significant part of solving problems with fractional exponents. When you see an exponent like \( \frac{1}{3} \), you are being asked to find the cube root of the number.
For \( \, -27 \, \), the cube root is \( -3 \). This is because \( (-3)^3 = -27 \). Knowing this, you can transform expressions involving cube roots into more straightforward numbers, leading directly into further calculations like raising to a power.
Steps to take a cube root:

• Identify the base, in this case \( -27 \).
• Calculate the value whose cube equals the base (\( -3 \) here).
• Confirm your cube root calculation by performing the cubic operation \( (-3) \, \times \, (-3) \, \times \, (-3) \).

This process simplifies exponential calculations, making it a vital skill for deeper comprehension of exponents.

Raising to a Power

After finding the root, the next step with fractional exponents is raising the result to a power.
When \( \, (-3)^{5} \, \) is calculated, it involves multiplying \( -3 \) by itself five times: \[ (-3) \times (-3) \times (-3) \times (-3) \times (-3) \]. This equals \( -243 \), since an odd number of negative factors results in a negative number.
Some things to remember when raising a number to a power:

• Be mindful of the sign of the base. An odd exponent results in a negative number if the base is negative.
• The process requires you to be precise with multiplication, ensuring each step is correctly done.
• This operation exponentially increases the number, which can be easily managed with solid arithmetic skills.

Combining this with root extraction complements your understanding of fractional exponents, closing the loop on exponent manipulation.