7) Simplify:a) $x^5 \cdot X^2 \cdot X^4$b) $c^2 D^7 \times C^5 D^2$c) $m^2 N \times M^3 N^2 P \times M \times N$d) $a^3 + A^3 \cdot A^3$e) $x Y Z \cdot X Y Z^4$

a) Simplify $x^5 \cdot x^2 \cdot x^4$

When simplifying algebraic expressions, we need to apply the rules of exponents. In this case, we have three terms with the same base, $x$. To simplify the expression, we need to multiply the exponents of $x$. The rule for multiplying exponents with the same base is to add the exponents. Therefore, we can simplify the expression as follows:

$x^5 \cdot x^2 \cdot x^4 = x^{5+2+4} = x^{11}$

This means that the simplified expression is $x^{11}$.

b) Simplify $c^2 d^7 \times c^5 d^2$

In this case, we have two terms with the same base, $c$, and two terms with the same base, $d$. To simplify the expression, we need to multiply the exponents of $c$ and $d$. The rule for multiplying exponents with the same base is to add the exponents. Therefore, we can simplify the expression as follows:

$c^2 d^7 \times c^5 d^2 = c^{2+5} d^{7+2} = c^7 d^9$

This means that the simplified expression is $c^7 d^9$.

c) Simplify $m^2 n \times m^3 n^2 p \times m \times n$

In this case, we have three terms with the same base, $m$, and three terms with the same base, $n$. We also have a term with the base $p$. To simplify the expression, we need to multiply the exponents of $m$ and $n$. The rule for multiplying exponents with the same base is to add the exponents. Therefore, we can simplify the expression as follows:

$m^2 n \times m^3 n^2 p \times m \times n = m^{2+3+1} n^{1+2+1} p = m^6 n^4 p$

This means that the simplified expression is $m^6 n^4 p$.

d) Simplify $a^3 + a^3 \cdot a^3$

In this case, we have two terms with the same base, $a$. To simplify the expression, we need to apply the rule for adding exponents with the same base. The rule is to add the exponents. Therefore, we can simplify the expression as follows:

$a^3 + a^3 \cdot a^3 = a^3 + a^{3+3} = a^3 + a^6$

This means that the simplified expression is $a^3 + a^6$.

e) Simplify $x y z \cdot x y z^4$

In this case, we have two terms with the same base, $x$, and two terms with the same base, $y$. We also have a term with the base $z$. To simplify the expression, we need to multiply the exponents of $x$ and $y$. The rule for multiplying exponents with the same base is to add the exponents. Therefore, we can simplify the expression as follows:

$x y z \cdot x y z^4 = x^{1+1} y^{1+1} z^{1+4} = x^2 y^2 z^5$

This means that the simplified expression is $x^2 y^2 z^5$.

Rules for Simplifying Algebraic Expressions

When simplifying algebraic expressions, we need to apply the rules of exponents. The rules for simplifying algebraic expressions are as follows:

• When multiplying exponents with the same base, add the exponents.
• When adding exponents with the same base, add the exponents.
• When multiplying exponents with different bases, multiply the exponents.
• When adding exponents with different bases, add the exponents.

Examples of Simplifying Algebraic Expressions

Here are some examples of simplifying algebraic expressions:

• $x^3 \cdot x^2 = x^{3+2} = x^5$
• $c^4 \cdot c^2 = c^{4+2} = c^6$
• $m^2 n \times m^3 n^2 p = m^{2+3} n^{1+2} p = m^5 n^3 p$
• $a^2 + a^2 \cdot a^2 = a^2 + a^{2+2} = a^2 + a^4$
• $x y z \cdot x y z^4 = x^{1+1} y^{1+1} z^{1+4} = x^2 y^2 z^5$

Conclusion

Simplifying algebraic expressions is an important concept in mathematics. By applying the rules of exponents, we can simplify complex expressions and make them easier to work with. In this article, we have discussed the rules for simplifying algebraic expressions and provided examples of how to simplify expressions using these rules. We have also discussed the importance of simplifying algebraic expressions and how it can be used in various mathematical applications.

Frequently Asked Questions

Q: What is the rule for multiplying exponents with the same base?

A: The rule for multiplying exponents with the same base is to add the exponents. For example, $x^3 \cdot x^2 = x^{3+2} = x^5$.

Q: What is the rule for adding exponents with the same base?

A: The rule for adding exponents with the same base is to add the exponents. For example, $a^2 + a^2 \cdot a^2 = a^2 + a^{2+2} = a^2 + a^4$.

Q: What is the rule for multiplying exponents with different bases?

A: The rule for multiplying exponents with different bases is to multiply the exponents. For example, $c^2 \cdot d^3 = c^2 \cdot d^3$.

Q: What is the rule for adding exponents with different bases?

A: The rule for adding exponents with different bases is to add the exponents. For example, $c^2 + d^3 = c^2 + d^3$.

Q: How do I simplify an expression with multiple terms?

A: To simplify an expression with multiple terms, you need to apply the rules of exponents to each term separately. For example, $m^2 n \times m^3 n^2 p \times m \times n = m^{2+3+1} n^{1+2+1} p = m^6 n^4 p$.

Q: What is the difference between multiplying and adding exponents?

A: Multiplying exponents involves adding the exponents when the bases are the same, while adding exponents involves adding the exponents when the bases are the same. For example, $x^3 \cdot x^2 = x^{3+2} = x^5$, while $a^2 + a^2 \cdot a^2 = a^2 + a^{2+2} = a^2 + a^4$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the rule that $a^{-n} = \frac{1}{a^n}$. For example, $x^{-2} = \frac{1}{x^2}$.

Q: What is the rule for simplifying expressions with fractional exponents?

A: The rule for simplifying expressions with fractional exponents is to rewrite the expression as a product of two terms, one with a positive exponent and the other with a negative exponent. For example, $x^{\frac{1}{2}} = \sqrt{x}$.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to apply the rules of exponents to each variable separately. For example, $x^2 y^3 \cdot x^4 y^2 = x^{2+4} y^{3+2} = x^6 y^5$.

Q: What is the difference between simplifying an expression and evaluating an expression?

A: Simplifying an expression involves rewriting the expression in a simpler form, while evaluating an expression involves finding the value of the expression. For example, $x^2 + 2x + 1$ is a simplified expression, while $x^2 + 2x + 1 = (x+1)^2$ is an evaluation of the expression.

Conclusion

Simplifying algebraic expressions is an important concept in mathematics. By applying the rules of exponents, we can simplify complex expressions and make them easier to work with. In this article, we have discussed the rules for simplifying algebraic expressions and provided examples of how to simplify expressions using these rules. We have also discussed frequently asked questions about simplifying algebraic expressions and provided answers to these questions.