Problem 12 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ 16^{2 / 3} \cdot 16^{-1 / 6} $$

Step-by-Step Solution

Answer

The value of the expression is 4.

Step 1: Understand the Meaning of Exponents

The expression \(16^{2/3} \cdot 16^{-1/6}\) involves powers of the same base, 16. Here, \(16^{2/3}\) means taking the cube root of 16, and then squaring the result. Similarly, \(16^{-1/6}\) means taking the sixth root of 16, then applying the negative exponent which will take the reciprocal of the result.

Step 2: Use Properties of Exponents

When multiplying like bases, we can add their exponents. So, combine the exponents of 16: \[ 16^{2/3} \cdot 16^{-1/6} = 16^{(2/3) + (-1/6)} \]

Step 3: Find a Common Denominator

To add the exponents \(2/3\) and \(-1/6\), find a common denominator. The common denominator of 3 and 6 is 6. Convert the fractions: \[ 2/3 = 4/6 \] The exponent \(-1/6\) is already in terms of sixths.

Step 4: Add the Exponents

Add the exponents with a common denominator: \[ 4/6 + (-1/6) = 3/6 = 1/2 \]

Step 5: Simplify the Expression

Now that the exponent is summed up, simplify the expression: \[ 16^{1/2} = \sqrt{16} = 4 \] The square root of 16 is 4.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is key to simplifying expressions effectively. Exponents represent how many times a base number is multiplied by itself. There are several important properties to remember:

• **Product of Powers Property**: When multiplying two powers with the same base, like in the expression \(16^{2/3} \cdot 16^{-1/6}\), you add the exponents. This is expressed as \(a^m \cdot a^n = a^{m+n}\).
• **Power of a Power Property**: If you raise an exponent to another power, you multiply the exponents, \((a^m)^n = a^{m \cdot n}\).
• **Negative Exponents**: A negative exponent means you take the reciprocal of the base, \(a^{-m} = \frac{1}{a^m}\).
• **Zero Exponents**: Any base raised to the power of zero is one, \(a^0 = 1\) (given that \(a eq 0\)).

Combining these properties helps solve and simplify expressions by understanding how to effectively manipulate and operate on exponents.

Simplifying Expressions

To simplify expressions involving exponents, like in the procedure shown, the primary goal is to reduce the expression to its simplest form using known properties.

• Start by identifying like bases. This helps in combining the terms by adding or subtracting the exponents as required.
• When adding fractions, as seen with \(\frac{2}{3}\) and \(-\frac{1}{6}\), it's vital to find a common denominator to proceed. Here, converting \(\frac{2}{3}\) to \(\frac{4}{6}\) smooths the process of combining the terms.
• After combining, check if the expression can be simplified further by applying roots or other simplification techniques.

In the given example, the expression reduces to a simple whole number, which makes it easy to interpret and apply.

Roots and Radicals

Roots and radicals are concepts closely related to exponents. The expression \(16^{1/2}\) in the last step is a great example of its application. Here’s how to navigate such expressions:

• **Square Roots**: Represented as \(a^{1/2}\), or \(\sqrt{a}\), it implies finding a number which, when multiplied by itself, gives \(a\).
• **Cube Roots and Beyond**: Exponents like \(a^{1/3}\) involve finding a number which, when used as a factor three times, gives \(a\). Generally, \(a^{1/n}\) signifies the nth root of \(a\).
• **Fractional Exponents**: They combine roots and powers. The numerator of a fractional exponent is the power, and the denominator is the root. With \(16^{2/3}\), you take the cube root of 16 and then square the result.

Understanding roots and radicals helps in simplifying complex expressions and in solving various mathematical problems efficiently.