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

Q: Why can't I just split 10a + 5x into 10a and 5x?

You can! That would give $ g^{10a} \times g^{5x} $, which is also correct. The given answer $ g^{5a+5x} \times g^{5a} $ is just one way to factor the expression.

Q: How do I know which way to split the exponent?

Look for common factors in the terms! Since 10a + 5x has a common factor of 5, you can write it as 5(2a + x). Then you can split it as (5a + 5x) + 5a or other combinations that add up correctly.

Q: What's the point of expanding if I can leave it as g^(10a+5x)?

Expanding helps you factor expressions and simplify calculations. Sometimes one factor might cancel out with other terms in a larger problem, making the math much easier!

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

Yes! For $ g^{5a+5x} \times g^{5a} $, add the exponents: (5a + 5x) + 5a = 5a + 5x + 5a = 10a + 5x ✓

Exponent Properties with Algebraic Factoring

Expand the following equation:

$g^{10a+5x}=$

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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, multiplication of exponents with the same base (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

00:17 We'll maintain the base and add the exponents together

00:20 We can observe that this expression is not equal to the original expression

00:23 We will use the same method in order to simplify the remaining expressions

00:26 In this expression the operation is addition and not multiplication, therefore it's not relevant

00:39 This expression is equal to the original expression

00:54 This expression is not 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:

$g^{10a+5x}=$

Step-by-step solution

To solve this problem, we will use the properties of exponents:

Step 1: Recognize that the exponent $10a + 5x$ can be expressed as the sum of two terms, namely $(5a + 5x) + 5a$ .
Step 2: Apply the property $g^{m+n} = g^{m} \times g^{n}$ to the expression $g^{10a + 5x}$ .
Step 3: Expand the expression using the identified split:
Since $10a + 5x = (5a + 5x) + 5a$ , we have:
$g^{10a + 5x} = g^{(5a + 5x) + 5a} = g^{5a + 5x} \times g^{5a}$ .

Thus, the expanded form of the expression is given by $g^{5a+5x} \times g^{5a}$ .

Final Answer

$g^{5a+5x}\times g^{5a}$

Key Points to Remember

Essential concepts to master this topic

Rule: Use $g^{m+n} = g^m \times g^n$ to split exponents
Technique: Factor 10a + 5x = (5a + 5x) + 5a before expanding
Check: Multiply back: $g^{5a+5x} \times g^{5a} = g^{10a+5x}$ ✓

Common Mistakes

Avoid these frequent errors

Randomly splitting the exponent without factoring
Don't split 10a + 5x into random parts like 5a + (2 + 5x) = wrong factors! This creates incorrect terms that don't add up to the original exponent. Always factor the exponent systematically by finding common terms before applying the exponent rule.

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 split 10a + 5x into 10a and 5x?

You can! That would give $g^{10a} \times g^{5x}$ , which is also correct. The given answer $g^{5a+5x} \times g^{5a}$ is just one way to factor the expression.

How do I know which way to split the exponent?

Look for common factors in the terms! Since 10a + 5x has a common factor of 5, you can write it as 5(2a + x). Then you can split it as (5a + 5x) + 5a or other combinations that add up correctly.

What's the point of expanding if I can leave it as g^(10a+5x)?

Expanding helps you factor expressions and simplify calculations. Sometimes one factor might cancel out with other terms in a larger problem, making the math much easier!

Can I check my answer by adding the exponents back?

Yes! For $g^{5a+5x} \times g^{5a}$ , add the exponents: (5a + 5x) + 5a = 5a + 5x + 5a = 10a + 5x ✓

What if the numbers don't factor nicely?

That's okay! Not all expressions factor perfectly. Sometimes you'll need to use different techniques or leave the expression in its simplest form without forcing a factorization.

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