Simplify t^6 × t^7: Exponent Multiplication Practice

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

When you multiply $ t^6 \times t^7 $, you're really multiplying 6 t's times 7 t's. That gives you a total of 6 + 7 = 13 t's multiplied together, which equals $ t^{13} $!

Q: What's the difference between this and raising a power to a power?

Great question! $ t^6 \times t^7 $ means multiply two powers (add exponents). But $ (t^6)^7 $ means raise a power to a power (multiply exponents: 6 × 7 = 42).

Q: Does this rule work with different bases?

No! This rule only works when the bases are exactly the same. You cannot simplify $ x^3 \times y^4 $ because x and y are different bases.

Q: Can I use this rule backwards to factor expressions?

Absolutely! If you see $ t^{13} $, you can write it as $ t^6 \times t^7 $ or $ t^5 \times t^8 $ or many other combinations.

Exponent Laws with Same Base Multiplication

Reduce the following equation:

$t^6\times t^7=$

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

00:07 equals the same base raised to the sum of the exponents (N+M)

00:12 We will apply this formula to our exercise

00:16 We'll maintain the base and add the exponents together

00:20 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:

$t^6\times t^7=$

Step-by-step solution

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

Step 1: Identify the base and the exponents from the given expression.
Step 2: Apply the rule for multiplying powers with the same base.
Step 3: Perform the addition to simplify the exponents.

Now, let's work through each step:
Step 1: The expression given is $t^6 \times t^7$ . The base here is $t$ , and the exponents are 6 and 7.

Step 2: According to the rule for multiplying exponents with the same base, we add the exponents. Therefore, the expression becomes:
$t^6 \times t^7 = t^{6+7}$ .

Step 3: Simplify the expression by adding the exponents:
$6 + 7 = 13$ .

Therefore, the simplified expression is $t^{13}$ .

The correct choice given is: $t^{13}$ , which is choice 2.

Final Answer

$t^{13}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $t^6 \times t^7 = t^{6+7} = t^{13}$
Check: Count total factors: t·t·t·t·t·t × t·t·t·t·t·t·t = 13 factors ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 6 × 7 = 42 to get $t^{42}$ ! This confuses the power rule with the multiplication rule and gives a completely wrong answer. 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?

When you multiply $t^6 \times t^7$ , you're really multiplying 6 t's times 7 t's. That gives you a total of 6 + 7 = 13 t's multiplied together, which equals $t^{13}$ !

What's the difference between this and raising a power to a power?

Great question! $t^6 \times t^7$ means multiply two powers (add exponents). But $(t^6)^7$ means raise a power to a power (multiply exponents: 6 × 7 = 42).

Does this rule work with different bases?

No! This rule only works when the bases are exactly the same. You cannot simplify $x^3 \times y^4$ because x and y are different bases.

What if one of the exponents is negative?

The rule still works! Just add the exponents as usual. For example: $t^6 \times t^{-2} = t^{6+(-2)} = t^4$ .

Can I use this rule backwards to factor expressions?

Absolutely! If you see $t^{13}$ , you can write it as $t^6 \times t^7$ or $t^5 \times t^8$ or many other combinations.

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