Solve x^2 × x^5: Multiplying Exponential Expressions

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

Because exponents show repeated multiplication! $ x^2 $ means x·x and $ x^5 $ means x·x·x·x·x. When you multiply them together, you get x·x·x·x·x·x·x, which is $ x^7 $.

Q: What if the bases are different, like $ x^2 \times y^5 $ ?

You cannot combine them! The rule only works when the bases are identical. $ x^2 \times y^5 $ stays as $ x^2y^5 $ - no simplification possible.

Q: Does this work with negative exponents too?

Absolutely! The rule works for all exponents. For example: $ x^{-3} \times x^5 = x^{-3+5} = x^2 $. Just add the exponents normally, even with negatives.

Q: How can I remember this rule easily?

Think: "Same base? Add the powers!" You can also remember that multiplication means "putting together" all the x's, so you're counting the total number of x's being multiplied.

Q: What's the difference between $ x^2 \times x^5 $ and $ (x^2)^5 $ ?

Big difference! $ x^2 \times x^5 = x^7 $ (add exponents), but $ (x^2)^5 = x^{10} $ (multiply exponents). The parentheses show power of a power, which uses a different rule.

Exponent Laws with Same Base Multiplication

$x^2\times x^5=$

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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

00:06 The power of the result equals the sum of the powers

00:09 We'll apply this formula to our exercise and add the powers together

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^2\times x^5=$

Step-by-step solution

Here we will have to to multiply terms with identical bases, therefore we use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$ Note that this property can only be used to calculate the multiplication between terms with identical bases,

From now on we no longer write the multiplication sign, but use the accepted form of writing in which placing terms next to each other means multiplication.

We apply it in the problem:

$x^2x^5=x^{2+5}=x^7$ Therefore, the correct answer is D.

Final Answer

$x^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $x^2 \times x^5 = x^{2+5} = x^7$
Check: Verify by expanding: x·x·x·x·x·x·x matches $x^7$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 2 × 5 = 10 to get $x^{10}$ ! This confuses the power rule with multiplication rule and gives a completely wrong answer. Always add exponents when multiplying terms with the same base: $x^2 \times x^5 = x^{2+5} = x^7$ .

Practice Quiz

Test your knowledge with interactive questions

$$

Simplify the following equation:

$5^8\times5^3=$

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

Because exponents show repeated multiplication! $x^2$ means x·x and $x^5$ means x·x·x·x·x. When you multiply them together, you get x·x·x·x·x·x·x, which is $x^7$ .

What if the bases are different, like $x^2 \times y^5$ ?

You cannot combine them! The rule only works when the bases are identical. $x^2 \times y^5$ stays as $x^2y^5$ - no simplification possible.

Does this work with negative exponents too?

Absolutely! The rule works for all exponents. For example: $x^{-3} \times x^5 = x^{-3+5} = x^2$ . Just add the exponents normally, even with negatives.

How can I remember this rule easily?

Think: "Same base? Add the powers!" You can also remember that multiplication means "putting together" all the x's, so you're counting the total number of x's being multiplied.

What's the difference between $x^2 \times x^5$ and $(x^2)^5$ ?

Big difference! $x^2 \times x^5 = x^7$ (add exponents), but $(x^2)^5 = x^{10}$ (multiply exponents). The parentheses show power of a power, which uses a different rule.

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Solve x^2 × x^5: Multiplying Exponential Expressions

Exponent Laws with Same Base Multiplication

❤️ 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 we add the exponents instead of multiplying them?

What if the bases are different, like $x^2 \times y^5$ ?

Does this work with negative exponents too?

How can I remember this rule easily?

What's the difference between $x^2 \times x^5$ and $(x^2)^5$ ?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Simplify a¹² ÷ a⁹ × a³ ÷ a⁴: Exponent Manipulation Practice

Simplify y³/y⁶ × y⁴/y⁻² × y¹²/y⁷: Complete Exponent Operation

Multiply and Simplify: b^(-3) × b^3 × b^4 × b^(-2) Expression

Simplify the Expression: a¹⁰ × b⁵ × a⁻² × b³

Solve x^2 × x^5: Multiplying Exponential Expressions

Exponent Laws with Same Base Multiplication

❤️ 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 we add the exponents instead of multiplying them?

What if the bases are different, like x2×y5 x^2 \times y^5 x2×y5?

Does this work with negative exponents too?

How can I remember this rule easily?

What's the difference between x2×x5 x^2 \times x^5 x2×x5 and (x2)5 (x^2)^5 (x2)5?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Simplify a¹² ÷ a⁹ × a³ ÷ a⁴: Exponent Manipulation Practice

Simplify y³/y⁶ × y⁴/y⁻² × y¹²/y⁷: Complete Exponent Operation

Multiply and Simplify: b^(-3) × b^3 × b^4 × b^(-2) Expression

Simplify the Expression: a¹⁰ × b⁵ × a⁻² × b³

What if the bases are different, like $x^2 \times y^5$ ?

What's the difference between $x^2 \times x^5$ and $(x^2)^5$ ?