Determine The Value Of The Variable In The Product Of Powers.1. What Is The Value Of X X X In The Product Of Powers 5 5 ⋅ 5 X = 5 2 5^5 \cdot 5^x = 5^2 5 5 ⋅ 5 X = 5 2 ? - $x = \square$2. What Is The Value Of Y Y Y In The Product Of $p^{-3}

Understanding the Product of Powers

The product of powers is a mathematical concept that states when we multiply two or more powers with the same base, we can add the exponents. This concept is essential in simplifying complex expressions and solving equations involving exponents. In this article, we will focus on determining the value of variables in the product of powers.

Solving for x in the Product of Powers $5^5 \cdot 5^x = 5^2$

To solve for x, we need to apply the product of powers rule, which states that when we multiply two or more powers with the same base, we can add the exponents. In this case, the base is 5, and the exponents are 5 and x.

Step 1: Apply the Product of Powers Rule

The product of powers rule states that when we multiply two or more powers with the same base, we can add the exponents. In this case, we can add the exponents 5 and x.

5^5 \cdot 5^x = 5^{5+x}

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents.

5+x = 2

Step 3: Solve for x

To solve for x, we need to isolate the variable x. We can do this by subtracting 5 from both sides of the equation.

x = 2 - 5
x = -3

Therefore, the value of x in the product of powers $5^5 \cdot 5^x = 5^2$ is -3.

Solving for y in the Product of Powers $p^{-3} \cdot p^y = p^4$

To solve for y, we need to apply the product of powers rule, which states that when we multiply two or more powers with the same base, we can add the exponents. In this case, the base is p, and the exponents are -3 and y.

Step 1: Apply the Product of Powers Rule

The product of powers rule states that when we multiply two or more powers with the same base, we can add the exponents. In this case, we can add the exponents -3 and y.

p^{-3} \cdot p^y = p^{-3+y}

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents.

-3+y = 4

Step 3: Solve for y

To solve for y, we need to isolate the variable y. We can do this by adding 3 to both sides of the equation.

y = 4 + 3
y = 7

Therefore, the value of y in the product of powers $p^{-3} \cdot p^y = p^4$ is 7.

Conclusion

In this article, we have learned how to determine the value of variables in the product of powers. We have applied the product of powers rule to solve for x in the product of powers $5^5 \cdot 5^x = 5^2$ and y in the product of powers $p^{-3} \cdot p^y = p^4$. We have also learned how to equate the exponents and solve for the variables. With this knowledge, we can now solve more complex equations involving exponents.

Key Takeaways

• The product of powers rule states that when we multiply two or more powers with the same base, we can add the exponents.
• To solve for a variable in the product of powers, we need to apply the product of powers rule and equate the exponents.
• We can solve for a variable by isolating the variable and solving for its value.

Practice Problems

1. What is the value of x in the product of powers $2^3 \cdot 2^x = 2^5$?
2. What is the value of y in the product of powers $q^{-2} \cdot q^y = q^3$?

Answer Key

1. x = 2
2. y = 5
   Product of Powers Q&A =========================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the product of powers.

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two or more powers with the same base, we can add the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, we need to identify the base and the exponents in the expression. We can then add the exponents and write the result as a single power with the same base.

Q: What if the exponents are negative?

A: If the exponents are negative, we can still apply the product of powers rule. We need to remember that a negative exponent means that the base is in the denominator.

Q: Can I apply the product of powers rule to expressions with different bases?

A: No, the product of powers rule only applies to expressions with the same base. If the bases are different, we need to use a different rule to simplify the expression.

Q: How do I solve for a variable in the product of powers?

A: To solve for a variable in the product of powers, we need to apply the product of powers rule and equate the exponents. We can then solve for the variable by isolating it and solving for its value.

Q: What if I have an expression with multiple variables?

A: If you have an expression with multiple variables, you can still apply the product of powers rule. You need to identify the base and the exponents in the expression, and then add the exponents. You can then solve for the variables by isolating them and solving for their values.

Q: Can I use the product of powers rule to simplify expressions with fractions?

A: Yes, you can use the product of powers rule to simplify expressions with fractions. You need to remember that a fraction can be written as a power with a negative exponent.

Q: How do I know when to use the product of powers rule?

A: You should use the product of powers rule when you have an expression with multiple powers with the same base. You can identify the base and the exponents in the expression, and then add the exponents.

Product of Powers Examples

Here are some examples of how to apply the product of powers rule:

Example 1

Simplify the expression $2^3 \cdot 2^4$ using the product of powers rule.

2^3 \cdot 2^4 = 2^{3+4} = 2^7

Example 2

Solve for x in the expression $3^x \cdot 3^2 = 3^5$ using the product of powers rule.

3^x \cdot 3^2 = 3^{x+2} = 3^5

Example 3

Simplify the expression $4^{-2} \cdot 4^3$ using the product of powers rule.

4^{-2} \cdot 4^3 = 4^{-2+3} = 4^1

Conclusion

In this article, we have answered some of the most frequently asked questions about the product of powers. We have also provided examples of how to apply the product of powers rule to simplify expressions and solve for variables. With this knowledge, you can now use the product of powers rule to simplify complex expressions and solve for variables.

Key Takeaways

• The product of powers rule states that when we multiply two or more powers with the same base, we can add the exponents.
• To apply the product of powers rule, we need to identify the base and the exponents in the expression.
• We can solve for a variable in the product of powers by applying the product of powers rule and equating the exponents.

Practice Problems

1. Simplify the expression $2^4 \cdot 2^3$ using the product of powers rule.
2. Solve for x in the expression $5^x \cdot 5^2 = 5^4$ using the product of powers rule.
3. Simplify the expression $3^{-2} \cdot 3^4$ using the product of powers rule.

Answer Key

1. $2^7$
2. x = 2
3. $3^2$