Solve x^□ = x×x×x×x×x×x: Finding the Missing Exponent

Q: How do I count the factors without making mistakes?

Point to each factor as you count: first x, second x, third x... Or use your fingers! For $ x \cdot x \cdot x \cdot x \cdot x \cdot x $, you should count six separate x's.

Q: What's the difference between the base and the exponent?

The base is the number being multiplied (x in this case). The exponent is the small number that tells you how many times to multiply the base by itself.

Q: Why isn't the answer 5 if I see 5 multiplication signs?

Great observation! You're counting the multiplication signs, but you need to count the factors. Five multiplication signs connect six factors: x·x·x·x·x·x has 5 dots but 6 x's!

Q: How can I check if my exponent is right?

Expand your answer back! If you think it's $ x^6 $, write out x·x·x·x·x·x and count. You should get the same number of factors as in the original problem.

Q: Does this work with any variable, not just x?

Absolutely! Whether it's $ a \cdot a \cdot a $ = $ a^3 $ or $ y \cdot y \cdot y \cdot y $ = $ y^4 $, the rule is the same: count the factors to find the exponent.

Exponent Rules with Repeated Multiplication

Fill in the missing number:

$x^☐=x\cdot x\cdot x\cdot x\cdot x\cdot x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing number

00:03 We will use the power formula

00:06 Any number (X) raised to any power (N)

00:09 equals X multiplied by itself N times

00:19 We will use this formula in our exercise

00:22 X is the number being multiplied

00:26 The number of multiplications equals the exponent (N)

00:32 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing number:

$x^☐=x\cdot x\cdot x\cdot x\cdot x\cdot x$

Step-by-step solution

To solve this problem, we'll adopt the following method:

Step 1: Understand the given expression in multiplication form: $x \cdot x \cdot x \cdot x \cdot x \cdot x$ .
Step 2: Count the number of times $x$ appears.
Step 3: Determine that the expression is equivalent to $x^n$ , where $n$ is the count of $x$ factors.

Now, let's work through these steps:

Step 1: We are given the product $x \cdot x \cdot x \cdot x \cdot x \cdot x$ , which is clearly a multiplication of $x$ six times.

Step 2: Counting these, we see that $x$ appears 6 times.

Step 3: Therefore, the product can be expressed as $x^6$ , meaning that the expression $x^☐ = x \cdot x \cdot x \cdot x \cdot x \cdot x$ implies that $☑ = 6$ .

Thus, the missing number is 6, corresponding to choice 2.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: An exponent tells how many times to multiply the base
Counting: Count each factor: $x \cdot x \cdot x \cdot x \cdot x \cdot x$ = 6 factors
Verification: Check by expanding $x^6$ back to six x's ✓

Common Mistakes

Avoid these frequent errors

Miscounting the number of factors
Don't count too quickly or miss factors = wrong exponent! Students often rush and count 5 instead of 6, giving $x^5$ instead of $x^6$ . Always count each factor carefully, pointing to each x as you count.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to the expression below?

$$ $$ 10,000^1 $$

FAQ

Everything you need to know about this question

How do I count the factors without making mistakes?

Point to each factor as you count: first x, second x, third x... Or use your fingers! For $x \cdot x \cdot x \cdot x \cdot x \cdot x$ , you should count six separate x's.

What's the difference between the base and the exponent?

The base is the number being multiplied (x in this case). The exponent is the small number that tells you how many times to multiply the base by itself.

Why isn't the answer 5 if I see 5 multiplication signs?

Great observation! You're counting the multiplication signs, but you need to count the factors. Five multiplication signs connect six factors: x·x·x·x·x·x has 5 dots but 6 x's!

How can I check if my exponent is right?

Expand your answer back! If you think it's $x^6$ , write out x·x·x·x·x·x and count. You should get the same number of factors as in the original problem.

Does this work with any variable, not just x?

Absolutely! Whether it's $a \cdot a \cdot a$ = $a^3$ or $y \cdot y \cdot y \cdot y$ = $y^4$ , the rule is the same: count the factors to find the exponent.

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