Problem 125 - 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.) $$ z^{\square} \cdot z^{\square}=z^{-10} $$

Step-by-Step Solution

Answer

Fill each box with -5.

Step 1: Identify the Base and Exponents

In the expression, we have the base as \(z\) and two exponents represented by boxes. This expression uses the property of exponents that states \(z^a \cdot z^b = z^{a+b}\). Our goal is to fill the squares such that the sum of the exponents equals -10.

Step 2: Set Up the Equation

To solve for the exponents, let's assume the exponents in the boxes are \(x\). Therefore, the equation becomes \(z^{x} \cdot z^{x} = z^{-10}\). According to the property of exponents, this simplifies to \(z^{x+x} = z^{-10}\), or \(z^{2x} = z^{-10}\).

Step 3: Solve for the Exponent

We can equate the exponents from \(z^{2x} = z^{-10}\) to solve for \(x\): \[ 2x = -10 \] Dividing both sides by 2 gives us:\[ x = \frac{-10}{2} = -5 \]

Step 4: Fill in the Boxes

Now that we have \(x = -5\), we can fill in both boxes with -5. Thus, the complete expression becomes \(z^{-5} \cdot z^{-5} = z^{-10}\). This satisfies the condition given in the problem.

Key Concepts

Properties of Exponents

Exponents are not just for making numbers look fancy — they follow specific rules that make it easier to work with them in algebra. One of the key properties of exponents is that when you multiply two numbers with the same base, you add their exponents together. This is expressed as \( z^a \cdot z^b = z^{a+b} \). This is a powerful rule because it simplifies expressions and helps when solving equations. Another interesting property is the power of a power rule, where you multiply the exponents when you raise a power to another power, written as \( (z^a)^b = z^{a\cdot b} \). These properties ensure we can manipulate expressions comfortably, keeping track of bases and their corresponding exponents.

Solving Equations

Solving equations might sound complicated at first, but it's really about balance. You want to find a number that makes an equation true. With exponents involved, you rely on properties like multiplying exponents to help find that number. Let's say you have the equation \( z^{2x} = z^{-10} \). To solve for \( x \), you need to understand that if the bases (which are the same here, both are \( z \)) are equal, the exponents themselves must be equal too.

• Start by equating the exponents: \( 2x = -10 \).
• Divide both sides of the equation by 2 to solve for \( x \).

When you solve \( 2x = -10 \), you get \( x = -5 \). So, the solution is reassuringly straightforward: make the terms on each side identical by setting their exponents equal. Consistency in equations is key.

Algebraic Expressions

Algebraic expressions involving exponents might seem daunting, but they are simply combinations of numbers, variables, and operations. Take a simple expression: \( z^{-5} \cdot z^{-5} \). Here, you have the same base, \( z \), being multiplied together with exponents of \(-5\). It might help to think of these expressions as recipes for calculation, with exponents describing how many times to multiply the base by itself. When filling in boxes like in the original exercise, you consider the properties of exponents to satisfy conditions of the equation, ensuring wherever you multiply expressions, the exponents sum up correctly — in this case, to \(-10\). Always remember, an expression needs to be coherent, each part adding meaning to the whole. And with practice, manipulating algebraic expressions becomes second nature.