Problem 126 - Algebra A Combined Function

Question

Fill in the boxes so that each statement is true. (More than one answer may be possible for these exercises.) $$ \left(x^{\square}\right)^{\square}=x^{-15} $$

Step-by-Step Solution

Answer

Possible pairs: \((-1, 15), (1, -15), (-3, 5), (3, -5)\).

Step 1: Understand the expression

The given equation is \((x^{\square})^{\square}=x^{-15}\). We need to find numbers that can fill the squares.

Step 2: Apply the Law of Exponents

The expression \((x^a)^b\) can be rewritten using the power of a power rule for exponents: \((x^a)^b = x^{a \cdot b}\). This step gives us \(x^{a \cdot b} = x^{-15}\).

Step 3: Equate the exponents

Since the bases of the exponents are the same, we equate the exponents: \(a \cdot b = -15\).

Step 4: Find possible values for the exponents

We need to find pairs of integers \(a\) and \(b\) such that \(a \cdot b = -15\). Some possible pairs are: \((-1, 15), (1, -15), (-3, 5), (3, -5)\).

Step 5: Verify solutions

Check each pair to ensure it satisfies the equation. Each pair \((a, b)\) when plugged into \((x^a)^b\) results in \(x^{-15}\) confirming they are correct.

Key Concepts

Power of a Power Rule

When you encounter expressions like \((x^a)^b\), the power of a power rule helps simplify them. Instead of doing operations separately, this rule tells you to multiply the exponents together. The expression becomes \(x^{a \cdot b}\).

For instance, if you have \((x^2)^3\), applying the power of a power rule gives you \(x^{2 \cdot 3} = x^6\).

This rule is handy because it simplifies complex expressions quickly. You don't need to expand terms manually; just multiply the exponents and you're good to go!

When working with these rules, always remember:

• The base \(x\) must be the same for both parts of the expression.
• Multiplying the exponents gives the new exponent for that base.

This rule makes handling exponents in algebraic expressions straightforward.

Negative Exponents

Negative exponents can seem a bit odd at first, but they're easy to understand. A negative exponent, like \(x^{-n}\), is equivalent to the reciprocal of the base raised to the positive exponent: \(x^{-n} = \frac{1}{x^n}\).

This means that instead of multiplying \(x\) by itself \(n\) times, you're dividing 1 by \(x\) \(n\) times.

Here's a quick example: consider \(x^{-2}\). This means \(\frac{1}{x^2}\). In practical terms, it means that as the exponent gets more negative, the value gets closer to zero (but never negative itself).

Key points to remember:

• Negative exponents indicate division or reciprocals.
• Always convert negative exponents to positive by using the reciprocal method.

Understanding negative exponents can help solve algebraic expressions effectively.

Integer Exponentiation

Integer exponentiation involves raising a number to the power of a whole number, including both positive and negative integers. This concept forms the backbone of many algebraic operations.

For example, \(2^3\) means multiplying 2 by itself three times, resulting in 8. However, if we consider \(2^{-3}\), it is the reciprocal operation, \(\frac{1}{2^3} = \frac{1}{8}\).

Why does this matter in algebra? Because integer exponents allow for:

• Efficient simplification of expressions.
• Clear definitions of how to handle division and multiplication operations.

Whenever you're dealing with integers in exponentiation, ensure you apply rules consistently by understanding:

• Positive exponents are straightforward multiplication.
• Negative exponents involve reciprocals.

Mastering this concept aids in managing and solving various mathematical problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operation symbols. They are foundational in algebra and come in various forms.

For instance, consider the expression \(3x + 5\). It's composed of:

• A coefficient \(3\) multiplied by the variable \(x\).
• A constant term \(5\).

Here, the goal is often to simplify or solve for the variable. With the equation \((x^a)^b = x^{-15}\), you are tasked with finding possible values for \(a\) and \(b\) that satisfy this form of expression.

In practice:

• Identify independent terms and coefficients.
• Apply laws like the power of a power rule to simplify.

Understanding how to work with algebraic expressions is crucial for tackling problems in algebra and beyond, as they form the language and groundwork of higher mathematics.