Simplify. Express Your Answer Using Positive Exponents.$\[ \frac{\left(7 T^{-1}\right)\left(6 T^8\right)}{3 T} \\]\[$\square\$\]

Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will focus on simplifying expressions using positive exponents, which is a crucial skill in algebra and other branches of mathematics. To begin with, let's understand the rules of exponents, which will help us simplify expressions effectively.

The Product Rule: When multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$. This rule is essential in simplifying expressions with multiple terms.

The Power Rule: When raising a power to another power, we multiply the exponents. For instance, $(a^m)^n = a^{m \cdot n}$. This rule helps us simplify expressions with repeated powers.

The Quotient Rule: When dividing two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$. This rule is vital in simplifying expressions with fractions.

Simplifying the Given Expression

Now that we have a solid understanding of the rules of exponents, let's apply them to simplify the given expression:

$\frac{\left(7 t^{-1}\right)\left(6 t^8\right)}{3 t}$

To simplify this expression, we will follow the order of operations (PEMDAS):

1. Multiply the numerators: $7 t^{-1} \cdot 6 t^8 = 42 t^{-1+8} = 42 t^7$
2. Divide the numerator by the denominator: $\frac{42 t^7}{3 t} = 14 t^{7-1} = 14 t^6$

Therefore, the simplified expression is:

$14 t^6$

Additional Examples and Practice

To reinforce our understanding of simplifying expressions using positive exponents, let's consider a few more examples:

Example 1

Simplify the expression: $\frac{\left(2 x^3\right)\left(4 x^{-2}\right)}{x^2}$

Using the product rule, we multiply the numerators: $2 x^3 \cdot 4 x^{-2} = 8 x^{3-2} = 8 x^1 = 8 x$

Then, using the quotient rule, we divide the numerator by the denominator: $\frac{8 x}{x^2} = 8 x^{1-2} = 8 x^{-1}$

Therefore, the simplified expression is:

$8 x^{-1}$

Example 2

Simplify the expression: $\frac{\left(3 y^4\right)\left(2 y^{-3}\right)}{6 y^2}$

Using the product rule, we multiply the numerators: $3 y^4 \cdot 2 y^{-3} = 6 y^{4-3} = 6 y^1 = 6 y$

Then, using the quotient rule, we divide the numerator by the denominator: $\frac{6 y}{6 y^2} = 6 y^{1-2} = 6 y^{-1}$

Therefore, the simplified expression is:

$6 y^{-1}$

Conclusion

Simplifying expressions using positive exponents is a crucial skill in mathematics, and it requires a solid understanding of the rules of exponents. By applying the product rule, power rule, and quotient rule, we can simplify complex expressions and arrive at a more manageable form. In this article, we have demonstrated how to simplify expressions using positive exponents, and we have provided additional examples and practice exercises to reinforce our understanding. With practice and patience, you will become proficient in simplifying expressions using positive exponents and will be able to tackle more complex mathematical problems with confidence.

Frequently Asked Questions

Q: What is the product rule of exponents?

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

Q: What is the power rule of exponents?

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

Q: What is the quotient rule of exponents?

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

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

A: To simplify an expression with multiple terms, we apply the product rule by adding the exponents of the terms with the same base.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, we apply the quotient rule by subtracting the exponents of the terms with the same base.

Final Thoughts

Simplifying expressions using positive exponents is a fundamental skill in mathematics, and it requires practice and patience to master. By understanding the rules of exponents and applying them effectively, you will be able to simplify complex expressions and arrive at a more manageable form. Remember to always follow the order of operations (PEMDAS) and to apply the product rule, power rule, and quotient rule to simplify expressions using positive exponents. With time and practice, you will become proficient in simplifying expressions using positive exponents and will be able to tackle more complex mathematical problems with confidence.

Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will focus on simplifying expressions using positive exponents, which is a crucial skill in algebra and other branches of mathematics. To begin with, let's understand the rules of exponents, which will help us simplify expressions effectively.

The Product Rule: When multiplying two numbers with the same base, we add their exponents. For example, $a^m \cdot a^n = a^{m+n}$. This rule is essential in simplifying expressions with multiple terms.

The Power Rule: When raising a power to another power, we multiply the exponents. For instance, $(a^m)^n = a^{m \cdot n}$. This rule helps us simplify expressions with repeated powers.

The Quotient Rule: When dividing two numbers with the same base, we subtract their exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$. This rule is vital in simplifying expressions with fractions.

Simplifying the Given Expression

Now that we have a solid understanding of the rules of exponents, let's apply them to simplify the given expression:

$\frac{\left(7 t^{-1}\right)\left(6 t^8\right)}{3 t}$

To simplify this expression, we will follow the order of operations (PEMDAS):

1. Multiply the numerators: $7 t^{-1} \cdot 6 t^8 = 42 t^{-1+8} = 42 t^7$
2. Divide the numerator by the denominator: $\frac{42 t^7}{3 t} = 14 t^{7-1} = 14 t^6$

Therefore, the simplified expression is:

$14 t^6$

Q&A: Simplifying Expressions Using Positive Exponents

Q: What is the product rule of exponents?

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

Q: What is the power rule of exponents?

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

Q: What is the quotient rule of exponents?

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

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

A: To simplify an expression with multiple terms, we apply the product rule by adding the exponents of the terms with the same base.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, we apply the quotient rule by subtracting the exponents of the terms with the same base.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle negative exponents?

A: When simplifying an expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent by applying the product rule, power rule, and quotient rule. For example, $\frac{x^m}{x^n} = x^{m-n}$.

Additional Examples and Practice

To reinforce our understanding of simplifying expressions using positive exponents, let's consider a few more examples:

Example 1

Simplify the expression: $\frac{\left(2 x^3\right)\left(4 x^{-2}\right)}{x^2}$

Using the product rule, we multiply the numerators: $2 x^3 \cdot 4 x^{-2} = 8 x^{3-2} = 8 x^1 = 8 x$

Then, using the quotient rule, we divide the numerator by the denominator: $\frac{8 x}{x^2} = 8 x^{1-2} = 8 x^{-1}$

Therefore, the simplified expression is:

$8 x^{-1}$

Example 2

Simplify the expression: $\frac{\left(3 y^4\right)\left(2 y^{-3}\right)}{6 y^2}$

Using the product rule, we multiply the numerators: $3 y^4 \cdot 2 y^{-3} = 6 y^{4-3} = 6 y^1 = 6 y$

Then, using the quotient rule, we divide the numerator by the denominator: $\frac{6 y}{6 y^2} = 6 y^{1-2} = 6 y^{-1}$

Therefore, the simplified expression is:

$6 y^{-1}$

Conclusion

Simplifying expressions using positive exponents is a crucial skill in mathematics, and it requires a solid understanding of the rules of exponents. By applying the product rule, power rule, and quotient rule, we can simplify complex expressions and arrive at a more manageable form. In this article, we have demonstrated how to simplify expressions using positive exponents, and we have provided additional examples and practice exercises to reinforce our understanding. With practice and patience, you will become proficient in simplifying expressions using positive exponents and will be able to tackle more complex mathematical problems with confidence.

Final Thoughts

Simplifying expressions using positive exponents is a fundamental skill in mathematics, and it requires practice and patience to master. By understanding the rules of exponents and applying them effectively, you will be able to simplify complex expressions and arrive at a more manageable form. Remember to always follow the order of operations (PEMDAS) and to apply the product rule, power rule, and quotient rule to simplify expressions using positive exponents. With time and practice, you will become proficient in simplifying expressions using positive exponents and will be able to tackle more complex mathematical problems with confidence.