Simplify the Algebraic Expression: Calculating x^3 × x^2 × x^{-2} × x^4

Q: Why do we add exponents instead of multiplying them?

Because each exponent tells us how many times to multiply the base by itself. When we multiply $ x^3 \times x^2 $, we're really doing x·x·x times x·x, which gives us 5 total x's, or $ x^5 $.

Q: How can I remember this rule?

Think of it as counting total factors. $ x^3 $ means 3 x's, $ x^2 $ means 2 more x's, etc. When you multiply them together, you're counting all the x's total.

Q: Can I solve this problem in a different order?

Absolutely! You can group and add exponents in any order you want. Try (3 + 4) + (2 + (-2)) = 7 + 0 = 7. The commutative property makes this flexible.

Exponent Rules with Mixed Signs

$x^3x^2x^{-2}x^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When multiplying powers with equal bases

00:07 The power of the result equals the sum of the powers

00:10 We'll apply this formula to our exercise and add the powers together

00:17 This is the solution

00:18 Chapter Title

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^3x^2x^{-2}x^4=$

Step-by-step solution

To solve this problem, we'll apply the rules for multiplying exponents:

Step 1: Identify the exponents on the base $x$ : - $x^3$ has an exponent of 3. - $x^2$ has an exponent of 2. - $x^{-2}$ has an exponent of -2. - $x^4$ has an exponent of 4.
Step 2: Apply the multiplication rule for exponents, which states that when multiplying like bases you add the exponents:
Calculate the total exponent by summing all individual exponents: $x^{3+2+(-2)+4}$ .
Step 3: Simplify the sum of the exponents:

$3 + 2 - 2 + 4 = 7$

Thus, the expression simplifies to $x^7$ .

Therefore, the simplified result of the given expression is $x^7$ .

Upon evaluating the given choices, the correct answer choice is option 4: $x^7$ .

Final Answer

$x^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add all exponents together
Technique: Calculate step-by-step: 3 + 2 + (-2) + 4 = 7
Check: Count total factors: x·x·x·x·x·x·x = x⁷ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply 3 × 2 × (-2) × 4 = -48 to get x⁻⁴⁸! This treats exponents like regular numbers being multiplied together. Always add the exponents when multiplying same bases: 3 + 2 + (-2) + 4 = 7.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents instead of multiplying them?

Because each exponent tells us how many times to multiply the base by itself. When we multiply $x^3 \times x^2$ , we're really doing x·x·x times x·x, which gives us 5 total x's, or $x^5$ .

What happens with the negative exponent x⁻²?

Negative exponents work the same way! When adding exponents, -2 is just another number to add: 3 + 2 + (-2) + 4. The negative doesn't change the addition rule.

How can I remember this rule?

Think of it as counting total factors. $x^3$ means 3 x's, $x^2$ means 2 more x's, etc. When you multiply them together, you're counting all the x's total.

What if I got x⁹ as my answer?

You probably forgot about the negative exponent! Make sure to include all exponents in your addition: 3 + 2 + (-2) + 4. Missing the -2 would give you 3 + 2 + 4 = 9.

Can I solve this problem in a different order?

Absolutely! You can group and add exponents in any order you want. Try (3 + 4) + (2 + (-2)) = 7 + 0 = 7. The commutative property makes this flexible.

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