Which Expression Is Equivalent To $5^2 \cdot 7^2$?A. $(5 \cdot 7)^2$Simplify \$(5 \cdot 7)^2$[/tex\].$(5 \cdot 7)^2 = $A. $14^2$ B. \$25^2$[/tex\] C. $35^2$ D.

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Introduction

In mathematics, expressions can be simplified using various techniques. One such technique is the use of the distributive property, which allows us to expand expressions and simplify them. In this article, we will explore how to simplify the expression $5^2 \cdot 7^2$ using the distributive property.

Understanding the Distributive Property

The distributive property is a fundamental concept in mathematics that allows us to expand expressions and simplify them. It states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be applied to expressions with multiple terms, allowing us to simplify them by expanding and combining like terms.

Simplifying the Expression

To simplify the expression $5^2 \cdot 7^2$, we can use the distributive property to expand it. We can start by rewriting the expression as:

52⋅72=(5⋅7)25^2 \cdot 7^2 = (5 \cdot 7)^2

This is because the distributive property allows us to expand the expression by multiplying the terms inside the parentheses.

Expanding the Expression

Now that we have rewritten the expression as $(5 \cdot 7)^2$, we can expand it using the distributive property. We can start by multiplying the terms inside the parentheses:

(5⋅7)2=(5⋅7)(5⋅7)(5 \cdot 7)^2 = (5 \cdot 7)(5 \cdot 7)

This can be further simplified by multiplying the terms:

(5⋅7)(5⋅7)=5⋅7⋅5⋅7(5 \cdot 7)(5 \cdot 7) = 5 \cdot 7 \cdot 5 \cdot 7

Combining Like Terms

Now that we have expanded the expression, we can combine like terms to simplify it. We can start by combining the terms with the same variable:

5⋅7⋅5⋅7=(5⋅5)(7⋅7)5 \cdot 7 \cdot 5 \cdot 7 = (5 \cdot 5)(7 \cdot 7)

This can be further simplified by multiplying the terms:

(5⋅5)(7⋅7)=25⋅49(5 \cdot 5)(7 \cdot 7) = 25 \cdot 49

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by multiplying the terms:

25⋅49=122525 \cdot 49 = 1225

Therefore, the expression $5^2 \cdot 7^2$ is equivalent to $1225$.

Conclusion

In this article, we have explored how to simplify the expression $5^2 \cdot 7^2$ using the distributive property. We have rewritten the expression as $(5 \cdot 7)^2$, expanded it using the distributive property, and combined like terms to simplify it. The final simplified expression is $1225$.

Answer

The correct answer is:

A. $(5 \cdot 7)^2 = 1225$

Discussion

The distributive property is a fundamental concept in mathematics that allows us to expand expressions and simplify them. In this article, we have explored how to simplify the expression $5^2 \cdot 7^2$ using the distributive property. We have rewritten the expression as $(5 \cdot 7)^2$, expanded it using the distributive property, and combined like terms to simplify it. The final simplified expression is $1225$.

Additional Examples

Here are some additional examples of how to simplify expressions using the distributive property:

  • 32⋅42=(3⋅4)2=122=1443^2 \cdot 4^2 = (3 \cdot 4)^2 = 12^2 = 144

  • 22⋅52=(2⋅5)2=102=1002^2 \cdot 5^2 = (2 \cdot 5)^2 = 10^2 = 100

  • 42⋅62=(4⋅6)2=242=5764^2 \cdot 6^2 = (4 \cdot 6)^2 = 24^2 = 576

These examples demonstrate how to simplify expressions using the distributive property.

Final Thoughts

Introduction

In our previous article, we explored how to simplify the expression $5^2 \cdot 7^2$ using the distributive property. We rewrote the expression as $(5 \cdot 7)^2$, expanded it using the distributive property, and combined like terms to simplify it. In this article, we will answer some frequently asked questions about simplifying expressions with the distributive property.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that allows us to expand expressions and simplify them. It states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be applied to expressions with multiple terms, allowing us to simplify them by expanding and combining like terms.

Q: How do I apply the distributive property to simplify an expression?

A: To apply the distributive property, you can follow these steps:

  1. Rewrite the expression as a product of sums.
  2. Expand the expression using the distributive property.
  3. Combine like terms to simplify the expression.

For example, to simplify the expression $5^2 \cdot 7^2$, you can rewrite it as $(5 \cdot 7)^2$, expand it using the distributive property, and combine like terms to simplify it.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different properties in mathematics.

The distributive property states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

The commutative property states that for any numbers a and b, the following equation holds:

a + b = b + a

The distributive property is used to expand expressions and simplify them, while the commutative property is used to rearrange terms in an expression.

Q: Can I use the distributive property to simplify expressions with fractions?

A: Yes, you can use the distributive property to simplify expressions with fractions. However, you need to be careful when multiplying fractions, as the product of two fractions is the product of the numerators divided by the product of the denominators.

For example, to simplify the expression $\frac{1}{2} \cdot \frac{3}{4}$, you can rewrite it as $(\frac{1}{2} \cdot \frac{3}{4})$, expand it using the distributive property, and combine like terms to simplify it.

Q: Can I use the distributive property to simplify expressions with exponents?

A: Yes, you can use the distributive property to simplify expressions with exponents. However, you need to be careful when multiplying exponents, as the product of two exponents is the product of the bases raised to the power of the sum of the exponents.

For example, to simplify the expression $2^2 \cdot 3^2$, you can rewrite it as $(2 \cdot 3)^2$, expand it using the distributive property, and combine like terms to simplify it.

Q: What are some common mistakes to avoid when using the distributive property?

A: Here are some common mistakes to avoid when using the distributive property:

  • Not rewriting the expression as a product of sums before expanding it.
  • Not combining like terms after expanding the expression.
  • Not being careful when multiplying fractions or exponents.
  • Not checking the final simplified expression for errors.

By avoiding these common mistakes, you can ensure that you are using the distributive property correctly and simplifying expressions accurately.

Conclusion

In this article, we have answered some frequently asked questions about simplifying expressions with the distributive property. We have discussed how to apply the distributive property, the difference between the distributive property and the commutative property, and how to use the distributive property to simplify expressions with fractions and exponents. We have also discussed some common mistakes to avoid when using the distributive property. By following these tips and avoiding these common mistakes, you can become proficient in using the distributive property to simplify expressions and solve mathematical problems.