Problem 26 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ (-27)^{5 / 3} $$

Step-by-Step Solution

Answer

The value of \((-27)^{5/3}\) is \(-243\).

Step 1: Understand the Expression

The expression \[(-27)^{5/3}\] involves raising the number \(-27\) to the power of \(\frac{5}{3}\). This can be viewed as a two-part operation: first taking the cube root, and then raising to the fifth power.

Step 2: Simplify the Root Part

Calculate the cube root of \(-27\). We know that \[-3 \times -3 \times -3 = -27\] thus, the cube root of \(-27\) is \(-3\). So we replace the base with its cube root:\[(-27)^{1/3} = -3\].

Step 3: Apply the Remaining Exponent

Now, take the result from Step 2, which is \(-3\), and raise it to the power of 5. Compute \[(-3)^5\].

Step 4: Compute the Power

Calculate \[(-3)^5\] as follows:\((-3) \times (-3) = 9\), \[9 \times (-3) = -27\], \[-27 \times (-3) = 81\], \[81 \times (-3) = -243\].Thus, \((-3)^5 = -243\).

Step 5: Verify the Solution

Verify that the calculations are correct by confirming each step. Ensure that all arithmetic operations, including the cube root and subsequent exponentiation, were correctly followed. The calculated expression should accurately reflect the original given expression. Thus, \[(-27)^{5/3} = -243\].

Key Concepts

Rational Exponents

Rational exponents can often seem challenging, but they are simpler than they appear. When you see an expression like \((-27)^{5/3}\), the exponent \(\frac{5}{3}\) is what we refer to as a rational exponent. It combines both an exponent and a root in a single operation.
Rational exponents are notated as a fraction, where the numerator represents the power, and the denominator represents the root. In our example, \(\frac{5}{3}\) denotes the fifth power and the cube root.

• The denominator, \(3\), tells us that we need to calculate the cube root first.
• The numerator, \(5\), indicates that we raise the result to the fifth power.

Understanding rational exponents requires seeing these two processes as steps of a single mathematical operation. By breaking down the steps, we simplify complex calculations.

Cube Root

The cube root is a fundamental concept used when dealing with rational exponents, especially when the denominator is \(3\). The cube root of a number \(x\) is the value that, when multiplied by itself three times, gives \(x\) back. Using our problem as an example, we calculate the cube root of \(-27\):

• We need to find a number that when multiplied three times results in \(-27\).
• The solution is \(-3\) because \((-3) \times (-3) \times (-3) = -27\).

Similarly, whenever handling cube roots, remember that they accept negative numbers. Negative numbers can have real cube roots because multiplying three instances of a negative number results in a negative product. This feature is distinctive from square roots, which do not accept negatives without introducing complex numbers. For understanding cube roots clearly, practice makes perfect.

Negative Bases

Working with negative bases can confuse because they behave differently from positive ones, especially when raised to powers. In our problem, the base \(-27\) is negative. When raising a negative number to a power, remember:

• If the exponent is an odd number, like \(5\), the result will be negative because multiplying an odd number of negative factors results in a negative value.
• If the exponent were even, the outcome would be positive due to pairing negative factors producing positive results.

In calculating \((-3)^5\), notice how each multiplication step alternates between negative and positive before landing on \(-243\). Understanding the sign of your final result is crucial when dealing with both odd and even exponents. Keep track of your multiplications using a step-by-step approach to ensure accuracy. Such careful handling is key when negative bases and high-powered exponents come together, like in our example.