Simplify the Expression: 2^10 × 2^7 × 2^6 Using Laws of Exponents

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

The multiplication rule for exponents says $ a^m \cdot a^n = a^{m+n} $. You're combining repeated multiplication, so you add how many times the base is used. Multiplying exponents is for a different rule: $ (a^m)^n = a^{m \cdot n} $.

Q: What if the bases were different numbers?

You cannot combine terms with different bases using this rule. For example, $ 2^5 \cdot 3^4 $ stays as is - you can only add exponents when the bases are identical.

Q: How do I remember which operation to use with exponents?

Multiplication = add exponents, Division = subtract exponents, Power of a power = multiply exponents. Think: same base multiplication means you're adding more of the same factor!

Q: Can I work with more than three terms?

Absolutely! The rule works for any number of terms: $ 2^a \cdot 2^b \cdot 2^c \cdot 2^d = 2^{a+b+c+d} $. Just add all the exponents together when the bases match.

Q: What's the difference between 2^23 and other answer choices?

$ 2^{13} $ would mean you only added two exponents, $ 2^{19} $ suggests incorrect addition, and $ 2^{420} $ means you multiplied instead of added. Only $ 2^{23} $ correctly adds all three: 10 + 7 + 6.

Exponent Laws with Multiple Base Terms

$2^{10}\cdot2^7\cdot2^6=$

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Understand the problem

$2^{10}\cdot2^7\cdot2^6=$

Step-by-step solution

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$ When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms in the multiplication have the same base, so we will use the previous property:

$2^{10}\cdot2^7\cdot2^6=2^{10+7+6}=2^{23}$ Therefore, the correct answer is option c.

Final Answer

$2^{23}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $2^{10} \cdot 2^7 \cdot 2^6 = 2^{10+7+6} = 2^{23}$
Check: Count total exponent additions: 10 + 7 + 6 = 23 ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't calculate 10 × 7 × 6 = 420 to get $2^{420}$ ! This confuses the multiplication rule with the power rule. Always add exponents when multiplying terms with the same base: 10 + 7 + 6 = 23.

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?

The multiplication rule for exponents says $a^m \cdot a^n = a^{m+n}$ . You're combining repeated multiplication, so you add how many times the base is used. Multiplying exponents is for a different rule: $(a^m)^n = a^{m \cdot n}$ .

What if the bases were different numbers?

You cannot combine terms with different bases using this rule. For example, $2^5 \cdot 3^4$ stays as is - you can only add exponents when the bases are identical.

How do I remember which operation to use with exponents?

Multiplication = add exponents, Division = subtract exponents, Power of a power = multiply exponents. Think: same base multiplication means you're adding more of the same factor!

Can I work with more than three terms?

Absolutely! The rule works for any number of terms: $2^a \cdot 2^b \cdot 2^c \cdot 2^d = 2^{a+b+c+d}$ . Just add all the exponents together when the bases match.

What's the difference between 2^23 and other answer choices?

$2^{13}$ would mean you only added two exponents, $2^{19}$ suggests incorrect addition, and $2^{420}$ means you multiplied instead of added. Only $2^{23}$ correctly adds all three: 10 + 7 + 6.

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