Simplify the Expression: y^2 · y^3 · y^6 Using Exponent Rules

Q: Why do I add the exponents instead of multiplying them?

Because exponents represent repeated multiplication! $ y^2 $ means y·y, and $ y^3 $ means y·y·y. So $ y^2 \cdot y^3 $ = y·y·y·y·y = $ y^5 $, which is 2+3 exponents.

Q: What if the bases were different letters like x and y?

Then you cannot combine them! The rule only works with the same base. So $ x^2 \cdot y^3 $ stays as $ x^2y^3 $ - you can't add those exponents.

Q: Does this work with negative exponents too?

Yes! The rule works with any exponents - positive, negative, or even fractions. For example: $ y^{-2} \cdot y^5 = y^{-2+5} = y^3 $.

Q: How do I remember when to add vs multiply exponents?

Multiplication of powers: ADD exponents ($ y^2 \cdot y^3 = y^{2+3} $)Power of a power: MULTIPLY exponents ($ (y^2)^3 = y^{2\times3} $)

Q: Why are both answers B and C correct in this problem?

Because $ y^{2+3+6} $ shows the process (adding exponents) and $ y^{11} $ shows the final simplified result. Both represent the same mathematical expression!

Exponent Rules with Multiple Same Bases

Reduce the following equation:

$y^2\cdot y^3\cdot y^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of exponents with equal bases (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:12 We will apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$y^2\cdot y^3\cdot y^6=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the exponents in the expression $y^2 \cdot y^3 \cdot y^6$ .
Step 2: Apply the exponent rule by adding the exponents together.
Step 3: Simplify the combined exponents to find the final expression.

Now, let's work through each step:

Step 1: The exponents in the expression are 2, 3, and 6.

Step 2: According to the multiplication rule for powers with the same base, we have $y^2 \cdot y^3 \cdot y^6 = y^{2+3+6}$ .

Step 3: Calculate the sum of the exponents: $2 + 3 + 6 = 11$ .

Therefore, the simplified expression is $y^{11}$ .

Given the choices, the correct answers, by these computations, correspond to:

$y^{11}$
$y^{2+3+6}$

Hence, the correct answer to the problem is B+C are correct.

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $y^2 \cdot y^3 \cdot y^6 = y^{2+3+6} = y^{11}$
Check: Count total y's multiplied: y·y·y·y·y·y·y·y·y·y·y = 11 y's ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $y^{2\times3\times6} = y^{36}$ ! This gives a completely wrong answer because you're raising to the power of 36 instead of 11. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

Because exponents represent repeated multiplication! $y^2$ means y·y, and $y^3$ means y·y·y. So $y^2 \cdot y^3$ = y·y·y·y·y = $y^5$ , which is 2+3 exponents.

What if the bases were different letters like x and y?

Then you cannot combine them! The rule only works with the same base. So $x^2 \cdot y^3$ stays as $x^2y^3$ - you can't add those exponents.

Does this work with negative exponents too?

Yes! The rule works with any exponents - positive, negative, or even fractions. For example: $y^{-2} \cdot y^5 = y^{-2+5} = y^3$ .

How do I remember when to add vs multiply exponents?

Multiplication of powers: ADD exponents ( $y^2 \cdot y^3 = y^{2+3}$ )
Power of a power: MULTIPLY exponents ( $(y^2)^3 = y^{2\times3}$ )

Why are both answers B and C correct in this problem?

Because $y^{2+3+6}$ shows the process (adding exponents) and $y^{11}$ shows the final simplified result. Both represent the same mathematical expression!

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