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}

by ADMIN 243 views

Understanding Product of Powers

In mathematics, the product of powers is a fundamental concept that allows us to simplify expressions by combining powers with the same base. The product of powers rule states that when we multiply two powers with the same base, we can add the exponents. This rule is represented as:

a^m * a^n = a^(m+n)

where 'a' is the base and 'm' and 'n' are the exponents.

Solving for Variables in Product of Powers

In this article, we will focus on solving for variables in product of powers. We will use the product of powers rule to simplify expressions and solve for the unknown variables.

Solving for x in 5^5 * 5^x = 5^2

To solve for x, we can use the product of powers rule. We know that the base is 5 and the exponents are 5 and x. We can add the exponents to get:

5^5 * 5^x = 5^(5+x)

Now, we can equate this expression to 5^2:

5^(5+x) = 5^2

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

5+x = 2

Now, we can solve for x by subtracting 5 from both sides:

x = 2 - 5 x = -3

Therefore, the value of x is -3.

Solving for y in p^(-3) * p^y = p^(-2)

To solve for y, we can use the product of powers rule. We know that the base is p and the exponents are -3 and y. We can add the exponents to get:

p^(-3) * p^y = p^(-3+y)

Now, we can equate this expression to p^(-2):

p^(-3+y) = p^(-2)

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

-3+y = -2

Now, we can solve for y by adding 3 to both sides:

y = -2 + 3 y = 1

Therefore, the value of y is 1.

Solving for x in a^x * a^(-x) = a^0

To solve for x, we can use the product of powers rule. We know that the base is a and the exponents are x and -x. We can add the exponents to get:

a^x * a^(-x) = a^(x-x)

Now, we can equate this expression to a^0:

a^(x-x) = a^0

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

x-x = 0

Now, we can solve for x by adding x to both sides:

2x = 0

Now, we can solve for x by dividing both sides by 2:

x = 0

Therefore, the value of x is 0.

Solving for y in b^y * b^(-y) = b^0

To solve for y, we can use the product of powers rule. We know that the base is b and the exponents are y and -y. We can add the exponents to get:

b^y * b^(-y) = b^(y-y)

Now, we can equate this expression to b^0:

b^(y-y) = b^0

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

y-y = 0

Now, we can solve for y by adding y to both sides:

2y = 0

Now, we can solve for y by dividing both sides by 2:

y = 0

Therefore, the value of y is 0.

Solving for x in c^x * c^(-x) = c^(-2)

To solve for x, we can use the product of powers rule. We know that the base is c and the exponents are x and -x. We can add the exponents to get:

c^x * c^(-x) = c^(x-x)

Now, we can equate this expression to c^(-2):

c^(x-x) = c^(-2)

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

x-x = -2

Now, we can solve for x by adding x to both sides:

2x = -2

Now, we can solve for x by dividing both sides by 2:

x = -1

Therefore, the value of x is -1.

Solving for y in d^y * d^(-y) = d^(-2)

To solve for y, we can use the product of powers rule. We know that the base is d and the exponents are y and -y. We can add the exponents to get:

d^y * d^(-y) = d^(y-y)

Now, we can equate this expression to d^(-2):

d^(y-y) = d^(-2)

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

y-y = -2

Now, we can solve for y by adding y to both sides:

2y = -2

Now, we can solve for y by dividing both sides by 2:

y = -1

Therefore, the value of y is -1.

Conclusion

In this article, we have solved for variables in product of powers using the product of powers rule. We have used the rule to simplify expressions and solve for the unknown variables. We have also seen that the product of powers rule can be used to solve for variables in a variety of expressions, including those with negative exponents.

Final Answer

The final answer is:

  • x = -3
  • y = 1
  • x = 0
  • y = 0
  • x = -1
  • y = -1
    Product of Powers Q&A ==========================

Frequently Asked Questions

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

Q: What is the product of powers rule?

A: The product of powers rule is a mathematical rule that states that when we multiply two powers with the same base, we can add the exponents. This rule is represented as:

a^m * a^n = a^(m+n)

where 'a' is the base and 'm' and 'n' are the exponents.

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

A: To apply the product of powers rule, you need to identify the base and the exponents in the expression. Then, you can add the exponents to get the final result.

Q: What if the exponents are negative?

A: If the exponents are negative, you can still apply the product of powers rule. For example:

a^(-m) * a^n = a^(-m+n)

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

A: Yes, you can use the product of powers rule with fractions. For example:

(a/b)^m * (a/b)^n = (a/b)^(m+n)

Q: How do I simplify expressions using the product of powers rule?

A: To simplify expressions using the product of powers rule, you need to identify the base and the exponents in the expression. Then, you can add the exponents to get the final result.

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

A: If you have an expression with multiple bases, you need to apply the product of powers rule separately for each base. For example:

a^m * b^n * c^p = a^m * (b^n * c^p)

Q: Can I use the product of powers rule with exponents with different bases?

A: No, you cannot use the product of powers rule with exponents with different bases. For example:

a^m * b^n ≠ a^(m+n)

Q: How do I solve for variables in product of powers?

A: To solve for variables in product of powers, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a variable in the exponent?

A: If you have an expression with a variable in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with complex numbers?

A: Yes, you can use the product of powers rule with complex numbers. For example:

(a+bi)^m * (a+bi)^n = (a+bi)^(m+n)

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

A: To apply the product of powers rule with radicals, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a radical in the exponent?

A: If you have an expression with a radical in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with trigonometric functions?

A: Yes, you can use the product of powers rule with trigonometric functions. For example:

sin^m(x) * sin^n(x) = sin^(m+n)(x)

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

A: To apply the product of powers rule with logarithmic functions, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a logarithmic function in the exponent?

A: If you have an expression with a logarithmic function in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with exponential functions?

A: Yes, you can use the product of powers rule with exponential functions. For example:

e^m * e^n = e^(m+n)

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

A: To apply the product of powers rule with hyperbolic functions, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a hyperbolic function in the exponent?

A: If you have an expression with a hyperbolic function in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with inverse functions?

A: Yes, you can use the product of powers rule with inverse functions. For example:

f^(-1)(x) * f^m(x) = f^(-1+m)(x)

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

A: To apply the product of powers rule with composite functions, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a composite function in the exponent?

A: If you have an expression with a composite function in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with parametric functions?

A: Yes, you can use the product of powers rule with parametric functions. For example:

x^m * y^n = (x*y)^(m+n)

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

A: To apply the product of powers rule with implicit functions, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with an implicit function in the exponent?

A: If you have an expression with an implicit function in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with differential equations?

A: Yes, you can use the product of powers rule with differential equations. For example:

dy/dx = (x^m * y^n) / (x^p * y^q)

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

A: To apply the product of powers rule with integral equations, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with an integral function in the exponent?

A: If you have an expression with an integral function in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with differential operators?

A: Yes, you can use the product of powers rule with differential operators. For example:

D^m(x) * D^n(x) = D^(m+n)(x)

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

A: To apply the product of powers rule with integral operators, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with an integral operator in the exponent?

A: If you have an expression with an integral operator in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with Fourier transforms?

A: Yes, you can use the product of powers rule with Fourier transforms. For example:

F^m(x) * F^n(x) = F^(m+n)(x)

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

A: To apply the product of powers rule with Laplace transforms, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a Laplace transform in the exponent?

A: If you have an expression with a Laplace transform in the exponent, you need to use the product of powers rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: Can I use the product of powers rule with Mellin transforms?

A: Yes, you can use the product of powers rule with Mellin transforms. For example:

M^m(x) * M^n(x) = M^(m+n)(x)

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

A: To apply the product of powers rule with Hankel transforms, you need to use the rule to simplify the expression. Then, you can equate the exponents to solve for the variable.

Q: What if I have an expression with a Hankel transform in the exponent?

A: If you have an