Problem 36 - Algebra 2

Question

Simplify each number. $$16^{1.5}$$

Step-by-Step Solution

Answer

The simplified form of \(16^{1.5}\) is 64.

Step 1: Recognize Conversion of Decimal Exponent

From the properties of exponents, analyze that \(16^{1.5}\) can be converted to a form which includes a square root, since 1.5 = 1 + 0.5 = 1 + 1/2. So our exercise becomes \(16^{1+\frac{1}{2}}\)

Step 2: Apply the Law of Exponents

From the law of exponents, we know that \(a^{p+q} = a^p \cdot a^q\). We apply this law to \(16^{1+\frac{1}{2}}\) which simplifies to \(16^1 \cdot 16^{\frac{1}{2}}\)

Step 3: Compute the Power and Square Root

Calculate the value of \(16^1\) which stays 16. Next, concentrate on the term \(16^{\frac{1}{2}}\). This is equal to the square root of 16. The square root of 16 is 4. Thus our expression reduces to \(16 \cdot 4\)

Step 4: Calculate the final Result

Compute the multiplication \(16 \cdot 4\) to get the simplified form of \(16^{1.5}\), yielding 64.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial to simplifying expressions like \(16^{1.5}\). Exponents tell you how many times to multiply a base number by itself.

Some key properties include:

• Product of Powers Property: This allows us to add the exponents when multiplying like bases, i.e., \(a^m \times a^n = a^{m+n}\).
• Power of a Power Property: Here, you multiply the exponents: \((a^m)^n = a^{m\times n}\).
• Fractional Exponents: These express roots, where \(a^{\frac{m}{n}}\) means \(\sqrt[n]{a^m}\). A common scenario is \(a^{\frac{1}{n}} = \sqrt[n]{a}\).

For \(16^{1.5}\), the decimal 1.5 can be rewritten as \(1 + \frac{1}{2}\). This helps us apply the product of powers property to simplify the expression

as \(16^1 \cdot 16^{\frac{1}{2}}\). This showcases the power of breaking down exponents to make calculations simpler and provides clarity by using familiar properties.

Square Root Calculation

Square roots are the reverse operation of squaring a number. They give you a number which, when multiplied by itself, gives you the original number you started with.

For instance, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\). When we encountered the term \(16^{\frac{1}{2}}\) while simplifying \(16^{1.5}\), we recognized this as the square root of 16.

Using the property of fractional exponents, \(a^{\frac{1}{2}} = \sqrt{a}\). Therefore, \(16^{\frac{1}{2}} = 4\). Understanding square roots allows us to simplify expressions involving exponents by converting them into more manageable parts. This significantly reduces the complexity

of calculations. By becoming familiar with square roots, you enhance your toolkit for solving various mathematical problems quickly and accurately.

Simplification of Expressions

Simplifying mathematical expressions involves reducing them to their most basic form without changing their value. This often requires applying various mathematical properties and operations.

• In our example, the exponent \(16^{1.5}\) is broken down into more straightforward components \(16^1 \cdot 16^{\frac{1}{2}}\) by using the properties of exponents.
  
  
• Each component is then computed separately: \(16^1\) is simply 16, and \(16^{\frac{1}{2}}\) is calculated as the square root of 16, which is 4.
  
  
• Finally, these results are combined through multiplication, yielding 64.
  
  

The purpose of simplification is not just to provide a final number but to demonstrate how large, complex expressions can be broken down into simpler, more understandable parts.

This can make mathematical problems seem less daunting and help verify your solution by checking each step.