Expand the Expression: 3 Raised to Power (12+10+5)

Q: Why can I split the exponent when there's addition?

The exponent addition rule works both ways! Just like $ 3^{12} \times 3^{10} = 3^{12+10} $, you can also go backwards: $ 3^{12+10} = 3^{12} \times 3^{10} $.

Q: What if the exponent had subtraction instead?

The same rule applies! For example, $ 3^{12-5} = \frac{3^{12}}{3^5} $. Addition becomes multiplication, and subtraction becomes division.

Q: Do I need to calculate the final numerical answer?

Not necessarily! The expanded form $ 3^{12} \times 3^{10} \times 3^5 $ is often the desired answer, especially when dealing with very large numbers.

Q: Can I group the exponents differently?

Yes! You could write $ 3^{22} \times 3^5 $ or $ 3^{12} \times 3^{15} $ since 12+10+5 = 22+5 = 12+15. All are correct expansions!

Q: What if the base numbers were different?

If you had $ 2^{12} \times 3^{10} \times 5^5 $, you cannot combine them because the bases are different. This exponent rule only works with the same base.

Exponent Laws with Addition in Exponents

Expand the following equation:

$3^{12+10+5}=$

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Understand the problem

Expand the following equation:

$3^{12+10+5}=$

Step-by-step solution

To expand the equation $3^{12+10+5}$ , we will apply the rule of exponents that states: when you multiply powers with the same base, you can add the exponents. However, in this case, we are starting with a single term and want to represent it as a product of terms with the base being raised to each of the individual exponents given in the sum. Here’s a step-by-step explanation:

1. Start with the expression: $3^{12+10+5}$ .

2. Recognize that the exponents are added together. According to the property of exponents (Multiplication of Powers), we can express a single power with summed exponents as a product of powers:

3. Break down the exponents: $3^{12+10+5} = 3^{12} \times 3^{10} \times 3^5$ .

4. As seen from the explanation: $3^{12+10+5}$ is expanded to the product $3^{12} \times 3^{10} \times 3^5$ by expressing each part of the sum as an exponent with the base 3.

The final expanded form is therefore: $3^{12} \times 3^{10} \times 3^5$

Final Answer

$3^{12}\times3^{10}\times3^5$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: When bases are same, $a^{m+n} = a^m \times a^n$
Technique: Split sum in exponent: $3^{12+10+5} = 3^{12} \times 3^{10} \times 3^5$
Check: Verify by adding exponents back: 12 + 10 + 5 = 27 ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't change $3^{12+10+5}$ to $3^{12 \times 10 \times 5}$ = completely wrong value! This confuses the addition rule with multiplication rule for exponents. Always remember that addition in the exponent stays as addition when expanding to separate terms.

Practice Quiz

Test your knowledge with interactive questions

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Simplify the following equation:

$5^8\times5^3=$

FAQ

Everything you need to know about this question

Why can I split the exponent when there's addition?

The exponent addition rule works both ways! Just like $3^{12} \times 3^{10} = 3^{12+10}$ , you can also go backwards: $3^{12+10} = 3^{12} \times 3^{10}$ .

What if the exponent had subtraction instead?

The same rule applies! For example, $3^{12-5} = \frac{3^{12}}{3^5}$ . Addition becomes multiplication, and subtraction becomes division.

Do I need to calculate the final numerical answer?

Not necessarily! The expanded form $3^{12} \times 3^{10} \times 3^5$ is often the desired answer, especially when dealing with very large numbers.

Can I group the exponents differently?

Yes! You could write $3^{22} \times 3^5$ or $3^{12} \times 3^{15}$ since 12+10+5 = 22+5 = 12+15. All are correct expansions!

What if the base numbers were different?

If you had $2^{12} \times 3^{10} \times 5^5$ , you cannot combine them because the bases are different. This exponent rule only works with the same base.

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