Problem 15 - Algebra for College Students

Question

Simplify each numerical expression. \(2^{7} \cdot 2^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Recognize the Base

In the expression \(2^7 \cdot 2^{-3}\), both terms have the same base of 2. This allows us to apply the laws of exponents.

Step 2: Apply the Law of Exponents

According to the law of exponents, when multiplying like bases, you add the exponents. So, compute \(2^{7 + (-3)}\).

Step 3: Perform Addition of Exponents

Add the exponents: \(7 + (-3) = 4\). This results in the new expression \(2^4\).

Step 4: Calculate the Power of Two

Calculate \(2^4\) by multiplying 2 by itself 3 more times: \(2 \times 2 \times 2 \times 2 = 16\).

Key Concepts

Multiplying Exponents

When you multiply numbers with the same base, use the property of exponents that states: you add the exponents together. For example, consider the expression \(a^m \cdot a^n\). Here, both parts of the expression have the same base \(a\).
This lets us use the multiplying exponents rule, simplifying it to \(a^{m+n}\). In the specific case of \(2^7 \cdot 2^{-3}\), you add the exponents 7 and -3 together:

• Step 1: Recognize that both parts have the same base of 2.
• Step 2: Use the exponent rule: \(2^{7 + (-3)}\).

This simplifies the expression while retaining the same base, making complex calculations much more straightforward.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, while ensuring not to change their value. By minimizing complexity, expressions become more manageable and easier to interpret. For example, consider our exercise \(2^7 \cdot 2^{-3}\).

• Combine like terms based on their base, here it's the number 2.
• Apply the rule for multiplying exponents to reduce the expression: \(2^{7 + (-3)} = 2^4\).

This process reduces two terms into a single term with simplified exponents, making further calculations more efficient using basic arithmetic.

Power of a Number

The power of a number tells us how many times to multiply the number by itself. For example, the expression \(2^4\) means multiplying 2 by itself three more times:

• Step 1: Start with 2.
• Step 2: Multiply by 2: \(2 \times 2 = 4\).
• Step 3: Multiply the result by 2: \(4 \times 2 = 8\).
• Step 4: Multiply again by 2: \(8 \times 2 = 16\).

Using powers of a number helps make large multiplications or divisions simpler and easier to comprehend. It enables you to see the operation's magnitude without doing lengthy manual multiplication.