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Sagot :
To simplify the expression [tex]\(\frac{\left(a^3 a^6\right)^{\frac{1}{4}}}{a^2}\)[/tex], we can follow these steps:
1. Simplify the expression inside the parentheses:
[tex]\[ a^3 \cdot a^6 \][/tex]
Here, we use the property of exponents that states when multiplying like bases, we add the exponents. So,
[tex]\[ a^3 \cdot a^6 = a^{3+6} = a^9 \][/tex]
2. Rewrite the expression with the simplified base:
[tex]\[ \frac{\left(a^9\right)^{\frac{1}{4}}}{a^2} \][/tex]
3. Apply the power rule to the numerator:
The power rule for exponents states that [tex]\((a^m)^n = a^{mn}\)[/tex]. Therefore,
[tex]\[ \left(a^9\right)^{\frac{1}{4}} = a^{9 \cdot \frac{1}{4}} = a^{\frac{9}{4}} \][/tex]
4. Rewrite the expression with the simplified numerator:
[tex]\[ \frac{a^{\frac{9}{4}}}{a^2} \][/tex]
5. Simplify the division of exponents:
When dividing like bases, we subtract the exponents:
[tex]\[ a^{\frac{9}{4}} \div a^2 = a^{\frac{9}{4} - 2} \][/tex]
Convert 2 to a fraction with the same denominator as [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[ 2 = \frac{8}{4} \][/tex]
6. Subtract the exponents:
[tex]\[ \frac{9}{4} - \frac{8}{4} = \frac{1}{4} \][/tex]
7. Write the final simplified expression:
[tex]\[ a^{\frac{1}{4}} \][/tex]
Thus, the simplified expression is:
[tex]\[ a^{\frac{1}{4}} \][/tex]
1. Simplify the expression inside the parentheses:
[tex]\[ a^3 \cdot a^6 \][/tex]
Here, we use the property of exponents that states when multiplying like bases, we add the exponents. So,
[tex]\[ a^3 \cdot a^6 = a^{3+6} = a^9 \][/tex]
2. Rewrite the expression with the simplified base:
[tex]\[ \frac{\left(a^9\right)^{\frac{1}{4}}}{a^2} \][/tex]
3. Apply the power rule to the numerator:
The power rule for exponents states that [tex]\((a^m)^n = a^{mn}\)[/tex]. Therefore,
[tex]\[ \left(a^9\right)^{\frac{1}{4}} = a^{9 \cdot \frac{1}{4}} = a^{\frac{9}{4}} \][/tex]
4. Rewrite the expression with the simplified numerator:
[tex]\[ \frac{a^{\frac{9}{4}}}{a^2} \][/tex]
5. Simplify the division of exponents:
When dividing like bases, we subtract the exponents:
[tex]\[ a^{\frac{9}{4}} \div a^2 = a^{\frac{9}{4} - 2} \][/tex]
Convert 2 to a fraction with the same denominator as [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[ 2 = \frac{8}{4} \][/tex]
6. Subtract the exponents:
[tex]\[ \frac{9}{4} - \frac{8}{4} = \frac{1}{4} \][/tex]
7. Write the final simplified expression:
[tex]\[ a^{\frac{1}{4}} \][/tex]
Thus, the simplified expression is:
[tex]\[ a^{\frac{1}{4}} \][/tex]
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