Which Expressions Are Equivalent To 1 36 \frac{1}{36} 36 1 ? Check All That Apply.A. 3 − 6 3^{-6} 3 − 6 B. 6 − 2 6^{-2} 6 − 2 C. 6 3 6 5 \frac{6^3}{6^5} 6 5 6 3 D. 6 2 6 − 1 \frac{6^2}{6^{-1}} 6 − 1 6 2 E. 8 − 9 ⋅ 8 7 8^{-9} \cdot 8^7 8 − 9 ⋅ 8 7

Mar 11, 2025 by ADMIN 253 views

$**Equivalent Expressions to $\frac{1}{36}$**$

In mathematics, equivalent expressions are those that have the same value, even if they appear to be different. In this article, we will explore which expressions are equivalent to $\frac{1}{36}$ .

Understanding the Problem

To begin, let's understand the given expression $\frac{1}{36}$ . This fraction can be expressed as a decimal, which is approximately 0.02778. However, we are interested in finding equivalent expressions, not just decimal approximations.

A. $3^{-6}$

The first option is $3^{-6}$ . To evaluate this expression, we need to understand the concept of negative exponents. A negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. In this case, $3^{-6}$ is equal to $\frac{1}{3^6}$ .

Now, let's calculate $3^6$ . This is equal to $3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$ . Therefore, $3^{-6}$ is equal to $\frac{1}{729}$ , which is not equivalent to $\frac{1}{36}$ .

B. $6^{-2}$

The second option is $6^{-2}$ . Similar to the previous option, we need to understand the concept of negative exponents. $6^{-2}$ is equal to $\frac{1}{6^2}$ .

Now, let's calculate $6^2$ . This is equal to $6 \times 6 = 36$ . Therefore, $6^{-2}$ is equal to $\frac{1}{36}$ , which is equivalent to the given expression.

C. $\frac{6^3}{6^5}$

The third option is $\frac{6^3}{6^5}$ . To evaluate this expression, we need to apply the quotient rule of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ . In this case, $\frac{6^3}{6^5}$ is equal to $6^{3-5}$ , which simplifies to $6^{-2}$ .

As we previously discussed, $6^{-2}$ is equal to $\frac{1}{36}$ , which is equivalent to the given expression.

D. $\frac{6^2}{6^{-1}}$

The fourth option is $\frac{6^2}{6^{-1}}$ . To evaluate this expression, we need to apply the quotient rule of exponents. $\frac{6^2}{6^{-1}}$ is equal to $6^{2-(-1)}$ , which simplifies to $6^3$ .

Now, let's calculate $6^3$ . This is equal to $6 \times 6 \times 6 = 216$ . Therefore, $\frac{6^2}{6^{-1}}$ is equal to $216$ , which is not equivalent to $\frac{1}{36}$ .

E. $8^{-9} \cdot 8^7$

The fifth option is $8^{-9} \cdot 8^7$ . To evaluate this expression, we need to apply the product rule of exponents, which states that $a^m \cdot a^n = a^{m+n}$ . In this case, $8^{-9} \cdot 8^7$ is equal to $8^{-9+7}$ , which simplifies to $8^{-2}$ .

Now, let's calculate $8^{-2}$ . This is equal to $\frac{1}{8^2}$ , which is equal to $\frac{1}{64}$ . Therefore, $8^{-9} \cdot 8^7$ is equal to $\frac{1}{64}$ , which is not equivalent to $\frac{1}{36}$ .

Conclusion

In conclusion, the equivalent expressions to $\frac{1}{36}$ are:

$6^{-2}$
$\frac{6^3}{6^5}$

These expressions have the same value as $\frac{1}{36}$ , making them equivalent. The other options, $3^{-6}$ , $\frac{6^2}{6^{-1}}$ , and $8^{-9} \cdot 8^7$ , are not equivalent to $\frac{1}{36}$ .

Final Answer

The final answer is:

A. No
B. Yes
C. Yes
D. No
E. No
Frequently Asked Questions (FAQs) about Equivalent Expressions =============================================================

In the previous article, we explored which expressions are equivalent to $\frac{1}{36}$ . In this article, we will answer some frequently asked questions (FAQs) about equivalent expressions.

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, even if they appear to be different.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can use the following steps:

Simplify both expressions by applying the rules of exponents, such as the product rule and the quotient rule.
Compare the simplified expressions to see if they are the same.
If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common rules for equivalent expressions?

A: Some common rules for equivalent expressions include:

The product rule: $a^m \cdot a^n = a^{m+n}$
The quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
The power rule: $(a^m)^n = a^{m \cdot n}$

Q: How do I apply the product rule?

A: To apply the product rule, you need to multiply the exponents of the two expressions. For example, if you have $a^m \cdot a^n$ , you can simplify it to $a^{m+n}$ .

Q: How do I apply the quotient rule?

A: To apply the quotient rule, you need to subtract the exponents of the two expressions. For example, if you have $\frac{a^m}{a^n}$ , you can simplify it to $a^{m-n}$ .

Q: How do I apply the power rule?

A: To apply the power rule, you need to multiply the exponent of the expression by the new exponent. For example, if you have $(a^m)^n$ , you can simplify it to $a^{m \cdot n}$ .

Q: What are some examples of equivalent expressions?

A: Some examples of equivalent expressions include:

$a^m \cdot a^n = a^{m+n}$
$\frac{a^m}{a^n} = a^{m-n}$
$(a^m)^n = a^{m \cdot n}$
$a^{-m} = \frac{1}{a^m}$
$a^0 = 1$

Q: How do I determine if an expression is equivalent to a fraction?

A: To determine if an expression is equivalent to a fraction, you can use the following steps:

Simplify the expression by applying the rules of exponents.
Compare the simplified expression to the fraction to see if they are the same.
If the simplified expression is the same as the fraction, then the original expression is equivalent to the fraction.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

Not simplifying expressions before comparing them.
Not applying the rules of exponents correctly.
Not checking if the simplified expressions are the same.

Q: How do I practice working with equivalent expressions?

A: To practice working with equivalent expressions, you can try the following:

Simplify expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use online resources or worksheets to practice working with equivalent expressions.

Q: What are some real-world applications of equivalent expressions?

A: Some real-world applications of equivalent expressions include:

Simplifying complex mathematical expressions in physics and engineering.
Working with equivalent fractions in finance and economics.
Simplifying expressions in computer programming.

Q: How do I use equivalent expressions in real-world problems?

A: To use equivalent expressions in real-world problems, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in physics, finance, and computer programming.

Q: What are some common misconceptions about equivalent expressions?

A: Some common misconceptions about equivalent expressions include:

Thinking that equivalent expressions are always the same.
Thinking that equivalent expressions can only be used in simple mathematical problems.
Thinking that equivalent expressions are only used in mathematics.

Q: How do I overcome common misconceptions about equivalent expressions?

A: To overcome common misconceptions about equivalent expressions, you can try the following:

Read and understand the rules of exponents.
Practice working with equivalent expressions.
Use online resources or worksheets to practice working with equivalent expressions.

Q: What are some resources for learning more about equivalent expressions?

A: Some resources for learning more about equivalent expressions include:

Online tutorials and videos.
Math textbooks and workbooks.
Online worksheets and practice problems.
Math apps and software.

Q: How do I use online resources to learn more about equivalent expressions?

A: To use online resources to learn more about equivalent expressions, you can try the following:

Search for online tutorials and videos.
Read and understand the rules of exponents.
Practice working with equivalent expressions using online worksheets and practice problems.
Use online math apps and software to practice working with equivalent expressions.

Q: What are some tips for mastering equivalent expressions?

A: Some tips for mastering equivalent expressions include:

Practice working with equivalent expressions regularly.
Read and understand the rules of exponents.
Use online resources and worksheets to practice working with equivalent expressions.
Apply equivalent expressions to real-world problems.

Q: How do I apply equivalent expressions to real-world problems?

A: To apply equivalent expressions to real-world problems, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in physics, finance, and computer programming.

Q: What are some common challenges when working with equivalent expressions?

A: Some common challenges when working with equivalent expressions include:

Not simplifying expressions before comparing them.
Not applying the rules of exponents correctly.
Not checking if the simplified expressions are the same.

Q: How do I overcome common challenges when working with equivalent expressions?

A: To overcome common challenges when working with equivalent expressions, you can try the following:

Read and understand the rules of exponents.
Practice working with equivalent expressions regularly.
Use online resources and worksheets to practice working with equivalent expressions.
Apply equivalent expressions to real-world problems.

Q: What are some benefits of mastering equivalent expressions?

A: Some benefits of mastering equivalent expressions include:

Simplifying complex mathematical expressions.
Solving real-world problems in physics, finance, and computer programming.
Improving math skills and problem-solving abilities.

Q: How do I use equivalent expressions to solve real-world problems?

A: To use equivalent expressions to solve real-world problems, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in physics, finance, and computer programming.

Q: What are some common applications of equivalent expressions in real-world problems?

A: Some common applications of equivalent expressions in real-world problems include:

Simplifying complex mathematical expressions in physics and engineering.
Working with equivalent fractions in finance and economics.
Simplifying expressions in computer programming.

Q: How do I apply equivalent expressions to real-world problems in physics?

A: To apply equivalent expressions to real-world problems in physics, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in physics, such as calculating the trajectory of a projectile.

Q: How do I apply equivalent expressions to real-world problems in finance?

A: To apply equivalent expressions to real-world problems in finance, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in finance, such as calculating the interest rate on a loan.

Q: How do I apply equivalent expressions to real-world problems in computer programming?

A: To apply equivalent expressions to real-world problems in computer programming, you can try the following:

Simplify complex mathematical expressions by applying the rules of exponents.
Compare simplified expressions to see if they are the same.
Use equivalent expressions to solve real-world problems in computer programming, such as calculating the area of a rectangle.

Q: What are some common mistakes to avoid when applying equivalent expressions to real-world problems?

A: Some common mistakes to avoid when applying equivalent expressions to real-world problems include:

Not simplifying expressions before comparing them.
Not applying the rules of exponents correctly.
Not checking if the simplified expressions are the same.

Q: How do I overcome common mistakes when applying equivalent expressions to real-world problems?

A: To overcome common mistakes when applying equivalent expressions to real-world problems, you can try the following:

Read and understand the rules of exponents.
Practice working with equivalent expressions regularly.
Use online resources and worksheets to practice working with equivalent expressions.
Apply equivalent expressions to real-world problems.

Q: What are some benefits of applying equivalent expressions to real-world problems?

A: Some benefits of applying equivalent expressions to real-world problems include:

Simplifying complex mathematical expressions.
Solving real-world problems in physics, finance, and computer programming.
Improving math skills and problem-solving abilities.