Expand the Power: Solving 7^(2x+7) Step by Step

Q: How do I know where to split the exponent?

Look at the answer choices for clues! They show you need two factors with x, so split $ 2x+7 $ as $ x + (x+7) $ to create $ 7^x \times 7^{x+7} $.

Q: What's the difference between $ 7^{x+7} \times 7^2 $ and $ 7^x \times 7^{x+7} $ ?

The first splits $ 2x+7 $ incorrectly as $ (x+7) + 2 $, but that doesn't equal $ 2x+7 $! The correct split is $ x + (x+7) = 2x+7 $.

Q: Can I check my answer by adding the exponents back?

Yes! For $ 7^x \times 7^{x+7} $, add exponents: $ x + (x+7) = x + x + 7 = 2x + 7 $. This matches the original exponent!

Q: Why is the exponent rule $ a^{m+n} = a^m \times a^n $ true?

Think of it as repeated multiplication! $ a^{m+n} $ means multiply a by itself (m+n) times. This is the same as multiplying a by itself m times, then n more times: $ a^m \times a^n $.

Exponent Rules with Sum Expansion

Expand the following equation:

$7^{2x+7}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Identify which expressions are equal to the original expression

00:04 According to the laws of exponents, the multiplication of powers with the same base (A)

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

00:11 We will apply this formula to our exercise

00:14 We'll maintain the base and add the exponents together

00:18 We can observe that this expression is not equal to the original expression

00:27 We will use the same method in order to simplify the remaining expressions

00:31 In this expression the operation is addition and not multiplication, therefore it's not relevant

00:40 This expression is not equal to the original expression

00:55 This expression is equal to the original expression

01:00 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

$7^{2x+7}=$

Step-by-step solution

To tackle this problem, we need to expand the expression $7^{2x+7}$ using the properties of exponents:

Step 1: Identify the structure of the exponent in the form $2x + 7$ .
Step 2: Recognize this as a sum of two components: $2x$ and $7$ .
Step 3: Apply the exponent rule $a^{m+n} = a^m \times a^n$ to separate the expression into two distinct exponent terms.

Now, let's proceed with the solution:

Step 1: Given the exponent is $2x + 7$ , break it down into individual components:
$2x + 7 = x + x + 7$ .

Step 2: Applying the rule $a^{m+n} = a^m \times a^n$ :
$7^{2x+7} = 7^{x+x+7} = 7^x \times 7^{x+7}$ .

Therefore, the expanded form of the expression is $7^x \times 7^{x+7}$ .

Final Answer

$7^x\times7^{x+7}$

Key Points to Remember

Essential concepts to master this topic

Rule: Use $a^{m+n} = a^m \times a^n$ to split sum exponents
Technique: Break $2x+7$ into $x+x+7$ then apply rule twice
Check: Verify $7^x \times 7^{x+7} = 7^{x+(x+7)} = 7^{2x+7}$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly splitting the exponent sum
Don't break $2x+7$ into $2x$ and $7$ = $7^{2x} \times 7^7$ ! This misses the variable structure. Always recognize that $2x = x+x$ to properly separate terms.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why can't I just write $7^{2x} \times 7^7$ ?

While this uses the exponent rule correctly, it doesn't match any of the answer choices! The problem wants you to split $2x$ further into $x + x$ to get $7^x \times 7^{x+7}$ .

How do I know where to split the exponent?

Look at the answer choices for clues! They show you need two factors with x, so split $2x+7$ as $x + (x+7)$ to create $7^x \times 7^{x+7}$ .

What's the difference between $7^{x+7} \times 7^2$ and $7^x \times 7^{x+7}$ ?

The first splits $2x+7$ incorrectly as $(x+7) + 2$ , but that doesn't equal $2x+7$ ! The correct split is $x + (x+7) = 2x+7$ .

Can I check my answer by adding the exponents back?

Yes! For $7^x \times 7^{x+7}$ , add exponents: $x + (x+7) = x + x + 7 = 2x + 7$ . This matches the original exponent!

Why is the exponent rule $a^{m+n} = a^m \times a^n$ true?

Think of it as repeated multiplication! $a^{m+n}$ means multiply a by itself (m+n) times. This is the same as multiplying a by itself m times, then n more times: $a^m \times a^n$ .

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Expand the Power: Solving 7^(2x+7) Step by Step

Exponent Rules with Sum Expansion

❤️ 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 can't I just write $7^{2x} \times 7^7$ ?

How do I know where to split the exponent?

What's the difference between $7^{x+7} \times 7^2$ and $7^x \times 7^{x+7}$ ?

Can I check my answer by adding the exponents back?

Why is the exponent rule $a^{m+n} = a^m \times a^n$ true?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: Solving 2^(2a+a) Step by Step

Expand the Expression: 3^(2a+x+a) Step by Step

Expand the Expression: Breaking Down g^(10a+5x)

Expand the Expression: 4^(a+b+c) Using Power Properties

Expand the Power: Solving 7^(2x+7) Step by Step

Exponent Rules with Sum Expansion

❤️ 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 can't I just write 72x×77 7^{2x} \times 7^7 72x×77?

How do I know where to split the exponent?

What's the difference between 7x+7×72 7^{x+7} \times 7^2 7x+7×72 and 7x×7x+7 7^x \times 7^{x+7} 7x×7x+7?

Can I check my answer by adding the exponents back?

Why is the exponent rule am+n=am×an a^{m+n} = a^m \times a^n am+n=am×an true?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: Solving 2^(2a+a) Step by Step

Expand the Expression: 3^(2a+x+a) Step by Step

Expand the Expression: Breaking Down g^(10a+5x)

Expand the Expression: 4^(a+b+c) Using Power Properties

Why can't I just write $7^{2x} \times 7^7$ ?

What's the difference between $7^{x+7} \times 7^2$ and $7^x \times 7^{x+7}$ ?

Why is the exponent rule $a^{m+n} = a^m \times a^n$ true?