Value Comparison: Determining the Greater Magnitude

Q: Why do I multiply exponents in $ (y^4)^3 $ instead of adding them?

Because you're taking $ y^4 $ and multiplying it by itself 3 times: $ y^4 \times y^4 \times y^4 $. Using the product rule, this becomes $ y^{4+4+4} = y^{12} $.

Q: How do I know which expression has the greatest value?

First, simplify each expression to the form $ y^n $. Then compare the exponents - the largest exponent gives the greatest value (assuming y > 1).

Q: What's the difference between $ y^7 \times y^2 $ and $ (y^7)^2 $ ?

In $ y^7 \times y^2 $, you add exponents: $ y^{7+2} = y^9 $. In $ (y^7)^2 $, you multiply exponents: $ y^{7 \times 2} = y^{14} $. Very different results!

Q: Why does $ \frac{y^{11}}{y^4} $ become $ y^7 $ ?

When dividing powers with the same base, you subtract the exponents: $ \frac{y^{11}}{y^4} = y^{11-4} = y^7 $. This is the quotient rule for exponents.

Q: What if y is negative or a fraction?

The exponent rules still work the same way! However, comparing values becomes trickier. For this problem, we assume y > 1 so higher exponents mean greater values.

Exponent Rules with Power Comparisons

Which value is greater?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the greatest value

00:04 According to the laws of exponents, multiplication with the same base(A) with exponents N,M

00:06 will be equal to the same base(A) with exponent (N+M)

00:10 Let's apply it to the question

00:14 We'll use the formula and combine the exponents

00:17 According to the laws of exponents, raising the base(A) with exponent(M) to the power of(N)

00:20 will give us the base(A) with an exponent that is the product of the powers(N,M)

00:23 Let's apply it to the question

00:27 We'll use the formula and multiply the exponents

00:34 According to the laws of exponents, when dividing with the same base(A)

00:37 with different exponents(M,N) where M is in the numerator

00:40 we obtain the same base(A) with exponent(M-N)

00:43 Let's apply it to the question and subtract the exponents

00:47 Always subtract the denominator's exponent from the numerator's exponent

00:52 Let's go through the answers and find the largest one

01:02 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which value is greater?

Step-by-step solution

To solve this problem, we need to simplify and compare the given expressions.

Let's simplify each:

$y^7 \times y^2$ :
Using the product of powers rule, $y^7 \times y^2 = y^{7+2} = y^9$ .
$(y^4)^3$ :
Using the power of a power rule, $(y^4)^3 = y^{4 \times 3} = y^{12}$ .
$y^9$ :
This is already in its simplest form, $y^9$ .
$\frac{y^{11}}{y^4}$ :
Using the power of a quotient rule, $\frac{y^{11}}{y^4} = y^{11-4} = y^7$ .

Now that all the expressions are in the form $y^n$ , we can compare the exponents to see which is greatest: $y^9$ , $y^{12}$ , $y^9$ , and $y^7$ .

The expression with the highest power is $y^{12}$ , which corresponds to the choice $(y^4)^3$ .

Thus, the greater value among the choices is $(y^4)^3$ .

Final Answer

$(y^4)^3$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add exponents: $y^a \times y^b = y^{a+b}$
Power of Power: $(y^4)^3 = y^{4 \times 3} = y^{12}$ multiplies exponents together
Verification: Convert all expressions to $y^n$ form and compare exponents ✓

Common Mistakes

Avoid these frequent errors

Confusing multiplication and addition of exponents
Don't add exponents in $(y^4)^3$ to get $y^7$ ! Power of a power means multiply exponents, not add them. This gives completely wrong values. Always multiply exponents when raising a power to another power: $(y^4)^3 = y^{4 \times 3} = y^{12}$ .

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

$$ $4^5\times4^5=$

FAQ

Everything you need to know about this question

Why do I multiply exponents in $(y^4)^3$ instead of adding them?

Because you're taking $y^4$ and multiplying it by itself 3 times: $y^4 \times y^4 \times y^4$ . Using the product rule, this becomes $y^{4+4+4} = y^{12}$ .

How do I know which expression has the greatest value?

First, simplify each expression to the form $y^n$ . Then compare the exponents - the largest exponent gives the greatest value (assuming y > 1).

What's the difference between $y^7 \times y^2$ and $(y^7)^2$ ?

In $y^7 \times y^2$ , you add exponents: $y^{7+2} = y^9$ . In $(y^7)^2$ , you multiply exponents: $y^{7 \times 2} = y^{14}$ . Very different results!

Why does $\frac{y^{11}}{y^4}$ become $y^7$ ?

When dividing powers with the same base, you subtract the exponents: $\frac{y^{11}}{y^4} = y^{11-4} = y^7$ . This is the quotient rule for exponents.

What if y is negative or a fraction?

The exponent rules still work the same way! However, comparing values becomes trickier. For this problem, we assume y > 1 so higher exponents mean greater values.

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Value Comparison: Determining the Greater Magnitude

Exponent Rules with Power Comparisons

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I multiply exponents in (y4)3 (y^4)^3 (y4)3 instead of adding them?

How do I know which expression has the greatest value?

What's the difference between y7×y2 y^7 \times y^2 y7×y2 and (y7)2 (y^7)^2 (y7)2?

Why does y11y4 \frac{y^{11}}{y^4} y4y11​ become y7 y^7 y7?

What if y is negative or a fraction?

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Why do I multiply exponents in $(y^4)^3$ instead of adding them?

What's the difference between $y^7 \times y^2$ and $(y^7)^2$ ?

Why does $\frac{y^{11}}{y^4}$ become $y^7$ ?