Solve (a×b×c)^xa: Product of Variables with Variable Exponent

Q: Why does every variable get the same exponent?

When you have $ (a \times b \times c)^{xa} $, the power of a product rule says the exponent applies to each factor equally. Think of it like multiplying the expression by itself xa times!

Q: What if the exponent was just x instead of xa?

The same rule applies! $ (abc)^x = a^x \times b^x \times c^x $. The exponent, whether it's a number, variable, or expression like xa, always distributes to each factor.

Q: Do I need to simplify xa as an exponent?

No! Keep $ xa $ as is in your final answer. It's already in simplest form as a product of variables in the exponent position.

Q: How can I remember this rule?

Think: "The exponent is like a multiplier that affects everyone equally." Just like sharing pizza slices - if there are 3 people and each gets xa slices, everyone gets the same amount!

Q: What's the difference between (abc)^xa and abc^xa?

Huge difference! $ (abc)^{xa} $ means all variables get the exponent, but $ abc^{xa} $ means only c gets the exponent: $ a \times b \times c^{xa} $. Parentheses matter!

Exponent Rules with Product Bases

Solve the following equation : $\left(a\times b\times c\right)^{xa}=$

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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, a product raised to the power (N)

00:07 equals a product where each factor is raised to the same power (N)

00:12 We will apply this formula to our exercise

00:15 We'll break down the product into each factor separately and raise to power (N)

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation : $\left(a\times b\times c\right)^{xa}=$

Step-by-step solution

Let's solve the problem step by step:

We need to simplify the expression $(a \times b \times c)^{xa}$ .

According to the power of a product rule, when you raise a product to an exponent, you raise each factor in the product to the exponent. Thus,
$(a \times b \times c)^{xa} = a^{xa} \times b^{xa} \times c^{xa}$ .

The exponent $xa$ is distributed to each base inside the parentheses.

This means that each of the individual terms $a$ , $b$ , and $c$ is raised to the power of $xa$ .

Therefore, the simplified form of the given expression is $a^{xa} \times b^{xa} \times c^{xa}$ .

Final Answer

$a^{xa}\times b^{xa}\times c^{xa}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute the exponent to each factor in the product
Technique: $(abc)^{xa} = a^{xa} \times b^{xa} \times c^{xa}$
Check: Each variable gets the same exponent $xa$ applied ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to first variable
Don't apply the exponent xa only to variable a = $a^{xa} \times b \times c$ ! This ignores the power rule and gives an incomplete result. Always distribute the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does every variable get the same exponent?

When you have $(a \times b \times c)^{xa}$ , the power of a product rule says the exponent applies to each factor equally. Think of it like multiplying the expression by itself xa times!

What if the exponent was just x instead of xa?

The same rule applies! $(abc)^x = a^x \times b^x \times c^x$ . The exponent, whether it's a number, variable, or expression like xa, always distributes to each factor.

Do I need to simplify xa as an exponent?

No! Keep $xa$ as is in your final answer. It's already in simplest form as a product of variables in the exponent position.

How can I remember this rule?

Think: "The exponent is like a multiplier that affects everyone equally." Just like sharing pizza slices - if there are 3 people and each gets xa slices, everyone gets the same amount!

What's the difference between (abc)^xa and abc^xa?

Huge difference! $(abc)^{xa}$ means all variables get the exponent, but $abc^{xa}$ means only c gets the exponent: $a \times b \times c^{xa}$ . Parentheses matter!

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