Problem 42 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ b^{9 / 5} b^{8 / 5} $$

Step-by-Step Solution

Answer

The expression simplifies to \( b^{17/5} \).

Step 1: Identify the Same Base

In the expression \( b^{9/5} b^{8/5} \), both terms have the same base \( b \). This allows us to use the properties of exponents effectively.

Step 2: Apply the Product of Powers Property

The Product of Powers Property states that when multiplying two expressions with the same base, you add the exponents: \( b^m \times b^n = b^{m+n} \). Here, we compute \( b^{9/5 + 8/5} \).

Step 3: Add the Exponents

To simplify the expression, add the exponents: \( \frac{9}{5} + \frac{8}{5} = \frac{17}{5} \). The expression is now \( b^{17/5} \).

Step 4: Write the Final Result

Since the expression \( b^{17/5} \) already has positive exponents, no further simplification is necessary. Thus, the simplified expression is \( b^{17/5} \).

Key Concepts

Product of Powers Property

When working with exponents, one crucial concept to understand is the Product of Powers Property. This property simplifies the process of multiplying terms with the same base. The rule states:

• When you multiply two exponents with the same base, you keep the base and add the exponents together.
• The mathematical formula is: \( b^m \times b^n = b^{m+n} \).

Let's break it down with an example: suppose you have the expression \( b^{9/5} \times b^{8/5} \). According to the product of powers property, you just need to add the exponents \( \frac{9}{5} \) and \( \frac{8}{5} \). This gives you \( b^{(9/5) + (8/5)} = b^{17/5} \). This property makes calculations easier and is a foundational skill for algebraic manipulation. Understanding and being able to apply the product of powers property can aid you in tackling more advanced problems in algebra and beyond.

Same Base Rule

In the world of exponents, having the same base simplifies operations significantly. The Same Base Rule is a fundamental concept used to apply properties of exponents effectively. When the bases of two terms are the same, you can utilize specific properties like the Product of Powers, discussed earlier. For instance:

• In the expression \( b^{9/5} \times b^{8/5} \), the base \( b \) is present in both terms. Identifying the base as being consistent is crucial to applying the rules of exponents.
• This allows us to focus solely on the exponents when multiplying or dividing, simplifying the expression considerably.

Remember, the same base rule is what lets you move ahead with operations such as adding exponents, rather than multiplying or performing any other operation on them. Always check for the base before trying to apply exponent rules!

Positive Exponents

Positive exponents represent straightforward multiplication and scaling of numbers or expressions. An exponent indicates how many times a base is used as a factor in multiplication. When we say 'positive exponents,' we refer to exponents greater than zero.

• In our example \( b^{17/5} \), the exponent \( \frac{17}{5} \) is positive, reflecting a large multiplication factor without any complications that might arise with negative or zero exponents.
• Positive exponents make working with expressions more straightforward, as you avoid reciprocal operations required for negative exponents.

Keeping exponents positive, when simplifying expressions, ensures clarity and precision in communicating mathematical ideas. Remember to always provide your final answers in terms of positive exponents unless otherwise specified. This will make sure the expression is easy to interpret and works in your favor mathematically.