Simplify And Express Your Answer Using Positive Exponents.$\[8 F^{32} G^{-43} H^0 \cdot 2 F^{27} G^6 H^{-3} \cdot 5 F^{-79} G H^{-3}\\]

Understanding the Problem

When dealing with exponents, it's essential to simplify expressions and express them using positive exponents. This involves applying the rules of exponents, such as the product rule, power rule, and quotient rule, to rewrite the expression in a more manageable form. In this article, we will focus on simplifying the given expression using positive exponents.

The Given Expression

The given expression is:

${8 f^{32} g^{-43} h^0 \cdot 2 f^{27} g^6 h^{-3} \cdot 5 f^{-79} g h^{-3}}$

This expression involves three variables: $f$, $g$, and $h$, each raised to various powers. Our goal is to simplify this expression and express it using positive exponents.

Applying the Product Rule

The product rule states that when multiplying two or more expressions with the same base, we add the exponents. In this case, we can apply the product rule to combine the terms with the same base.

${8 f^{32} g^{-43} h^0 \cdot 2 f^{27} g^6 h^{-3} \cdot 5 f^{-79} g h^{-3}}$

Using the product rule, we can rewrite the expression as:

${16 f^{32+27-79} g^{-43+6+1} h^{-3}}$

Simplifying the exponents, we get:

${16 f^{0} g^{-36} h^{-3}}$

Applying the Power Rule

The power rule states that when raising a power to another power, we multiply the exponents. In this case, we can apply the power rule to simplify the expression further.

${16 f^{0} g^{-36} h^{-3}}$

Using the power rule, we can rewrite the expression as:

${16 (f^0)^1 (g^{-36})^1 (h^{-3})^1}$

Simplifying the expression, we get:

${16 f^0 g^{-36} h^{-3}}$

Simplifying the Expression

Now that we have applied the product rule and power rule, we can simplify the expression further by combining the terms with the same base.

${16 f^0 g^{-36} h^{-3}}$

Since $f^0 = 1$, we can rewrite the expression as:

${16 g^{-36} h^{-3}}$

Expressing the Answer Using Positive Exponents

Our goal is to express the answer using positive exponents. To do this, we can use the rule that states $a^{-n} = \frac{1}{a^n}$. Applying this rule, we can rewrite the expression as:

${\frac{16}{g^{36} h^3}}$

Conclusion

In this article, we simplified the given expression using positive exponents. We applied the product rule and power rule to combine the terms with the same base and simplify the expression. Finally, we expressed the answer using positive exponents by applying the rule that states $a^{-n} = \frac{1}{a^n}$. This demonstrates the importance of understanding and applying the rules of exponents in simplifying expressions and expressing them in a more manageable form.

Key Takeaways

• The product rule states that when multiplying two or more expressions with the same base, we add the exponents.
• The power rule states that when raising a power to another power, we multiply the exponents.
• To express an answer using positive exponents, we can use the rule that states $a^{-n} = \frac{1}{a^n}$.

Practice Problems

1. Simplify the expression: $2x^3y^{-2}z^4 \cdot 3x^2y^3z^{-1} \cdot 5x^{-1}y^2z^2$
2. Express the answer using positive exponents: $\frac{1}{a^3b^2c^4}$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Glossary

• Product Rule: When multiplying two or more expressions with the same base, we add the exponents.
• Power Rule: When raising a power to another power, we multiply the exponents.
• Positive Exponents: Exponents that are greater than or equal to 1.
  Simplify and Express Your Answer Using Positive Exponents: Q&A ================================================================

Understanding the Problem

When dealing with exponents, it's essential to simplify expressions and express them using positive exponents. This involves applying the rules of exponents, such as the product rule, power rule, and quotient rule, to rewrite the expression in a more manageable form. In this article, we will focus on simplifying the given expression using positive exponents and answer some frequently asked questions.

Q&A

Q: What is the product rule in exponents?

A: The product rule states that when multiplying two or more expressions with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the power rule in exponents?

A: The power rule states that when raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression using positive exponents?

A: To simplify an expression using positive exponents, you can apply the product rule and power rule to combine the terms with the same base. For example, $a^m \cdot a^n = a^{m+n}$ and $(a^m)^n = a^{m \cdot n}$.

Q: What is the difference between positive and negative exponents?

A: Positive exponents are greater than or equal to 1, while negative exponents are less than 1. For example, $a^2$ is a positive exponent, while $a^{-2}$ is a negative exponent.

Q: How do I express an answer using positive exponents?

A: To express an answer using positive exponents, you can use the rule that states $a^{-n} = \frac{1}{a^n}$. For example, $\frac{1}{a^2} = a^{-2}$.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables by applying the product rule and power rule to combine the terms with the same base. For example, $a^m \cdot b^n \cdot c^p = a^{m+n+p} \cdot b^n \cdot c^p$.

Q: What is the quotient rule in exponents?

A: The quotient rule states that when dividing two expressions with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I apply the quotient rule in exponents?

A: To apply the quotient rule in exponents, you can subtract the exponents of the two expressions with the same base. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Practice Problems

1. Simplify the expression: $2x^3y^{-2}z^4 \cdot 3x^2y^3z^{-1} \cdot 5x^{-1}y^2z^2$
2. Express the answer using positive exponents: $\frac{1}{a^3b^2c^4}$
3. Simplify the expression: $a^m \cdot b^n \cdot c^p$
4. Express the answer using positive exponents: $\frac{1}{a^2b^3c^4}$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Glossary

• Product Rule: When multiplying two or more expressions with the same base, we add the exponents.
• Power Rule: When raising a power to another power, we multiply the exponents.
• Positive Exponents: Exponents that are greater than or equal to 1.
• Negative Exponents: Exponents that are less than 1.
• Quotient Rule: When dividing two expressions with the same base, we subtract the exponents.