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Math  /  Algebra

QuestionSet FF 1.) ab4c3a4b7c2\frac{a b^{4} c^{3}}{a^{4} b^{7} c^{2}} 2.) −3m3⋅m5⋅2m-3 m^{3} \cdot m^{5} \cdot 2 m 3.) (5n44n2)2\left(\frac{5 n^{4}}{4 n^{2}}\right)^{2}

Studdy Solution

STEP 1

1. The given expressions involve simplifying algebraic fractions and products.
2. The simplified expressions will involve properties of exponents and basic arithmetic operations.

STEP 2

1. Simplify the first expression using the properties of exponents.
2. Simplify the second expression by combining like terms and multiplying the constants.
3. Simplify the third expression by first simplifying inside the parentheses and then applying the power to the result.

STEP 3

Simplify the first expression ab4c3a4b7c2\frac{a b^{4} c^{3}}{a^{4} b^{7} c^{2}} by using the properties of exponents (i.e., xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}).
ab4c3a4b7c2=a1−4b4−7c3−2=a−3b−3c1 \frac{a b^{4} c^{3}}{a^{4} b^{7} c^{2}} = a^{1-4} b^{4-7} c^{3-2} = a^{-3} b^{-3} c^{1}

STEP 4

Simplify the second expression −3m3⋅m5⋅2m-3 m^{3} \cdot m^{5} \cdot 2 m by combining the constants and using the properties of exponents (i.e., xm⋅xn=xm+nx^m \cdot x^n = x^{m+n}).
−3⋅2⋅m3+5+1=−6m9 -3 \cdot 2 \cdot m^{3+5+1} = -6 m^{9}

STEP 5

Simplify the third expression (5n44n2)2\left(\frac{5 n^{4}}{4 n^{2}}\right)^{2} by simplifying inside the parentheses first and then applying the power outside the parentheses.
(5n44n2)=54n4−2=54n2 \left(\frac{5 n^{4}}{4 n^{2}}\right) = \frac{5}{4} n^{4-2} = \frac{5}{4} n^{2}
Then, apply the power of 2 to the simplified fraction:
(54n2)2=(54)2⋅(n2)2=2516n4 \left(\frac{5}{4} n^{2}\right)^{2} = \left(\frac{5}{4}\right)^{2} \cdot (n^{2})^{2} = \frac{25}{16} n^{4}
Solution:
1. ab4c3a4b7c2=a−3b−3c\frac{a b^{4} c^{3}}{a^{4} b^{7} c^{2}} = a^{-3} b^{-3} c
2. −3m3⋅m5⋅2m=−6m9-3 m^{3} \cdot m^{5} \cdot 2 m = -6 m^{9}
3. (5n44n2)2=2516n4\left(\frac{5 n^{4}}{4 n^{2}}\right)^{2} = \frac{25}{16} n^{4}

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