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Answer :
To solve the problem, we're looking to fill in the blanks in the expression [tex]\(\sqrt{\square x^{\square}}=4 x^9\)[/tex].
To break it down step-by-step:
1. Understand the Goal: We want the expression inside the square root, when simplified, to equal [tex]\(4 x^9\)[/tex].
2. Square Root Property: Recall that [tex]\(\sqrt{A} = B\)[/tex] means [tex]\(A = B^2\)[/tex].
3. Apply to Variables with Exponents:
- [tex]\(\sqrt{a} = b\)[/tex] implies [tex]\(a = b^2\)[/tex].
- Similarly, [tex]\(\sqrt{x^{C}} = x^{D}\)[/tex] implies [tex]\(x^{C} = (x^{D})^2 = x^{2D}\)[/tex].
Now let's match the given expression [tex]\(\sqrt{\square x^{\square}}\)[/tex] to [tex]\(4 x^9\)[/tex]:
- Identify [tex]\(4 x^9\)[/tex]: The expression simplifies to:
- The number part is 4, and 4 squared is 16.
- The exponent on [tex]\(x\)[/tex] is 9, so solving [tex]\(x^{?}\)[/tex] gives us an exponent term of 18 as [tex]\((x^9)^2 = x^{18}\)[/tex].
Therefore:
4. Determine [tex]\(\square\)[/tex]:
- The square of 4 is 16. Hence, the number under the square root should be 16.
- Since we are looking for [tex]\((x^9)^2\)[/tex], this becomes [tex]\(x^{18}\)[/tex].
So, the complete expression under the square root should be:
- [tex]\(16\)[/tex] for the number,
- [tex]\(x^{18}\)[/tex] for the exponent.
Therefore, filling in the blanks, the expression should be [tex]\(\sqrt{16 x^{18}}\)[/tex].
In conclusion, the answer is:
- [tex]\(\boxed{16}\)[/tex] for the first blank,
- [tex]\(\boxed{18}\)[/tex] for the second blank.
To break it down step-by-step:
1. Understand the Goal: We want the expression inside the square root, when simplified, to equal [tex]\(4 x^9\)[/tex].
2. Square Root Property: Recall that [tex]\(\sqrt{A} = B\)[/tex] means [tex]\(A = B^2\)[/tex].
3. Apply to Variables with Exponents:
- [tex]\(\sqrt{a} = b\)[/tex] implies [tex]\(a = b^2\)[/tex].
- Similarly, [tex]\(\sqrt{x^{C}} = x^{D}\)[/tex] implies [tex]\(x^{C} = (x^{D})^2 = x^{2D}\)[/tex].
Now let's match the given expression [tex]\(\sqrt{\square x^{\square}}\)[/tex] to [tex]\(4 x^9\)[/tex]:
- Identify [tex]\(4 x^9\)[/tex]: The expression simplifies to:
- The number part is 4, and 4 squared is 16.
- The exponent on [tex]\(x\)[/tex] is 9, so solving [tex]\(x^{?}\)[/tex] gives us an exponent term of 18 as [tex]\((x^9)^2 = x^{18}\)[/tex].
Therefore:
4. Determine [tex]\(\square\)[/tex]:
- The square of 4 is 16. Hence, the number under the square root should be 16.
- Since we are looking for [tex]\((x^9)^2\)[/tex], this becomes [tex]\(x^{18}\)[/tex].
So, the complete expression under the square root should be:
- [tex]\(16\)[/tex] for the number,
- [tex]\(x^{18}\)[/tex] for the exponent.
Therefore, filling in the blanks, the expression should be [tex]\(\sqrt{16 x^{18}}\)[/tex].
In conclusion, the answer is:
- [tex]\(\boxed{16}\)[/tex] for the first blank,
- [tex]\(\boxed{18}\)[/tex] for the second blank.
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