Simplify The Expression:$5p^2 \cdot 3p^2$

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Understanding the Problem

When simplifying an expression, we need to combine like terms and apply the rules of exponents. In this case, we are given the expression $5p^2 \cdot 3p^2$ and we need to simplify it.

What are Like Terms?

Like terms are terms that have the same variable raised to the same power. In this case, both terms have the variable $p$ raised to the power of 2. Therefore, we can combine them.

Applying the Rules of Exponents

When multiplying two terms with the same base (in this case, $p$), we add the exponents. This is known as the product of powers rule.

Simplifying the Expression

Using the product of powers rule, we can simplify the expression as follows:

$5p^2 \cdot 3p^2 = 5 \cdot 3 \cdot p^2 \cdot p^2$

Combining the Coefficients

We can combine the coefficients (the numbers in front of the variables) by multiplying them together.

$5 \cdot 3 = 15$

Combining the Variables

We can combine the variables by adding their exponents.

$p^2 \cdot p^2 = p^{2+2} = p^4$

Simplifying the Expression

Now that we have combined the coefficients and the variables, we can simplify the expression as follows:

$5p^2 \cdot 3p^2 = 15p^4$

Conclusion

In this article, we have simplified the expression $5p^2 \cdot 3p^2$ by combining like terms and applying the rules of exponents. We have shown that the simplified expression is $15p^4$.

Frequently Asked Questions

• What are like terms? Like terms are terms that have the same variable raised to the same power.
• How do we simplify an expression with like terms? We combine the coefficients and add the exponents of the variables.
• What is the product of powers rule? The product of powers rule states that when multiplying two terms with the same base, we add the exponents.

Example Problems

• Simplify the expression $2x^2 \cdot 4x^2$ Using the product of powers rule, we can simplify the expression as follows:

$2x^2 \cdot 4x^2 = 2 \cdot 4 \cdot x^2 \cdot x^2$ $= 8 \cdot x^{2+2}$ $= 8x^4$

• Simplify the expression $3y^3 \cdot 2y^3$ Using the product of powers rule, we can simplify the expression as follows:

$3y^3 \cdot 2y^3 = 3 \cdot 2 \cdot y^3 \cdot y^3$ $= 6 \cdot y^{3+3}$ $= 6y^6$

Tips and Tricks

• When simplifying an expression, make sure to combine like terms and apply the rules of exponents.
• Use the product of powers rule to simplify expressions with like terms.
• Make sure to combine the coefficients and add the exponents of the variables.

Conclusion

In this article, we have simplified the expression $5p^2 \cdot 3p^2$ by combining like terms and applying the rules of exponents. We have shown that the simplified expression is $15p^4$. We have also provided example problems and tips and tricks to help you simplify expressions with like terms.

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Frequently Asked Questions

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. In the expression $5p^2 \cdot 3p^2$, both terms have the variable $p$ raised to the power of 2, so they are like terms.

Q: How do we simplify an expression with like terms?

A: To simplify an expression with like terms, we combine the coefficients (the numbers in front of the variables) by multiplying them together, and we add the exponents of the variables.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two terms with the same base, we add the exponents. In the expression $5p^2 \cdot 3p^2$, we have the same base ($p$) and the same exponent (2), so we add the exponents to get $p^{2+2} = p^4$.

Q: Can we simplify an expression with unlike terms?

A: No, we cannot simplify an expression with unlike terms. Unlike terms are terms that have the same variable but different exponents. For example, in the expression $5p^2 + 3p^3$, we cannot combine the terms because they have different exponents.

Q: How do we simplify an expression with variables and constants?

A: To simplify an expression with variables and constants, we can combine the variables by adding their exponents, and we can combine the constants by multiplying them together. For example, in the expression $5p^2 + 3p^2$, we can combine the variables to get $8p^2$.

Q: Can we simplify an expression with negative exponents?

A: Yes, we can simplify an expression with negative exponents. When we have a negative exponent, we can rewrite the expression with a positive exponent by taking the reciprocal of the base. For example, in the expression $5p^{-2}$, we can rewrite it as $\frac{5}{p^2}$.

Q: How do we simplify an expression with fractions?

A: To simplify an expression with fractions, we can multiply the numerator and denominator by the same value to eliminate the fraction. For example, in the expression $\frac{5p^2}{3p^2}$, we can multiply the numerator and denominator by $3p^2$ to get $5p^2 \cdot \frac{1}{3p^2} = \frac{5}{3}$.

Q: Can we simplify an expression with variables and fractions?

A: Yes, we can simplify an expression with variables and fractions. We can combine the variables by adding their exponents, and we can combine the fractions by multiplying the numerators and denominators together. For example, in the expression $\frac{5p^2}{3p^2} + \frac{2p^2}{4p^2}$, we can combine the variables to get $\frac{5}{3} + \frac{2}{4}$, and then we can combine the fractions to get $\frac{5 \cdot 4}{3 \cdot 4} + \frac{2 \cdot 3}{4 \cdot 3} = \frac{20}{12} + \frac{6}{12} = \frac{26}{12}$.

Example Problems

• Simplify the expression $2x^2 \cdot 4x^2$ Using the product of powers rule, we can simplify the expression as follows:

$2x^2 \cdot 4x^2 = 2 \cdot 4 \cdot x^2 \cdot x^2$ $= 8 \cdot x^{2+2}$ $= 8x^4$

• Simplify the expression $3y^3 \cdot 2y^3$ Using the product of powers rule, we can simplify the expression as follows:

$3y^3 \cdot 2y^3 = 3 \cdot 2 \cdot y^3 \cdot y^3$ $= 6 \cdot y^{3+3}$ $= 6y^6$

Tips and Tricks

• When simplifying an expression, make sure to combine like terms and apply the rules of exponents.
• Use the product of powers rule to simplify expressions with like terms.
• Make sure to combine the coefficients and add the exponents of the variables.
• When simplifying an expression with variables and fractions, make sure to combine the variables and fractions separately.

Conclusion

In this article, we have answered some frequently asked questions about simplifying expressions with like terms. We have also provided example problems and tips and tricks to help you simplify expressions with like terms.