Simplifying Algebraic Expressions With Exponents And Radicals

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This article delves into the simplification of complex algebraic expressions, focusing on those involving exponents, radicals, and fractions. We will meticulously break down the expression, explaining each step in detail to ensure clarity and understanding. The primary expression we aim to simplify is:

(16)n+2⋅23n4n⋅(xn)4xn+4⋅x3n+1⋅[94+916+65]1/2⋅[9+1625+23]1/2⋅[(23)−2+1]\frac{(\sqrt{16})^{n+2} \cdot 2^{3n}}{4^n} \cdot \frac{(x^n)^4}{x^{n+4}} \cdot x^{3n+1} \cdot \left[\frac{9}{4} + \frac{9}{16} + \frac{6}{5}\right]^{1/2} \cdot \left[9 + \frac{16}{25} + \frac{2}{3}\right]^{1/2} \cdot \left[\left(\frac{2}{3}\right)^{-2} + 1\right]

Step-by-Step Simplification

To effectively simplify this expression, we'll tackle it piece by piece. First, we'll address the terms involving numerical exponents, then move onto the variable exponents, and finally handle the radical terms.

Simplifying Numerical Exponents

Let's begin with the numerical part of the expression:

(16)n+2â‹…23n4n\frac{(\sqrt{16})^{n+2} \cdot 2^{3n}}{4^n}

Simplifying the Base: The square root of 16 is 4, so we can rewrite the expression as:

4n+2â‹…23n4n\frac{4^{n+2} \cdot 2^{3n}}{4^n}

Now, express 4 as 222^2:

(22)n+2â‹…23n(22)n\frac{(2^2)^{n+2} \cdot 2^{3n}}{(2^2)^n}

Using the power of a power rule, (ab)c=abc(a^b)^c = a^{bc}, we get:

22(n+2)â‹…23n22n\frac{2^{2(n+2)} \cdot 2^{3n}}{2^{2n}}

This simplifies to:

22n+4â‹…23n22n\frac{2^{2n+4} \cdot 2^{3n}}{2^{2n}}

When multiplying powers with the same base, we add the exponents. So, the numerator becomes:

22n+4+3n=25n+42^{2n+4+3n} = 2^{5n+4}

Now the expression looks like this:

25n+422n\frac{2^{5n+4}}{2^{2n}}

When dividing powers with the same base, we subtract the exponents:

2(5n+4)−2n=23n+42^{(5n+4) - 2n} = 2^{3n+4}

Thus, the simplified form of the numerical exponent part is:

23n+42^{3n+4}

Simplifying Variable Exponents

Next, let's simplify the terms involving the variable xx:

(xn)4xn+4â‹…x3n+1\frac{(x^n)^4}{x^{n+4}} \cdot x^{3n+1}

Applying the Power of a Power Rule: First, simplify (xn)4(x^n)^4 using the power of a power rule:

(xn)4=x4n(x^n)^4 = x^{4n}

Now the expression is:

x4nxn+4â‹…x3n+1\frac{x^{4n}}{x^{n+4}} \cdot x^{3n+1}

Dividing Powers with the Same Base: When dividing powers with the same base, subtract the exponents:

x4nxn+4=x4n−(n+4)=x4n−n−4=x3n−4\frac{x^{4n}}{x^{n+4}} = x^{4n - (n+4)} = x^{4n - n - 4} = x^{3n-4}

Now multiply by x3n+1x^{3n+1}:

x3n−4⋅x3n+1x^{3n-4} \cdot x^{3n+1}

Multiplying Powers with the Same Base: When multiplying powers with the same base, add the exponents:

x(3n−4)+(3n+1)=x6n−3x^{(3n-4) + (3n+1)} = x^{6n-3}

Therefore, the simplified form of the variable exponent part is:

x6n−3x^{6n-3}

Simplifying Radical Terms

Now, let's tackle the radical terms:

[94+916+65]1/2⋅[9+1625+23]1/2⋅[(23)−2+1]\left[\frac{9}{4} + \frac{9}{16} + \frac{6}{5}\right]^{1/2} \cdot \left[9 + \frac{16}{25} + \frac{2}{3}\right]^{1/2} \cdot \left[\left(\frac{2}{3}\right)^{-2} + 1\right]

Simplifying the First Radical Term: Let's simplify the first bracketed term:

94+916+65\frac{9}{4} + \frac{9}{16} + \frac{6}{5}

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4, 16, and 5 is 80. Convert each fraction to have this denominator:

94=9×204×20=18080\frac{9}{4} = \frac{9 \times 20}{4 \times 20} = \frac{180}{80}

916=9×516×5=4580\frac{9}{16} = \frac{9 \times 5}{16 \times 5} = \frac{45}{80}

65=6×165×16=9680\frac{6}{5} = \frac{6 \times 16}{5 \times 16} = \frac{96}{80}

Now add the fractions:

18080+4580+9680=180+45+9680=32180\frac{180}{80} + \frac{45}{80} + \frac{96}{80} = \frac{180 + 45 + 96}{80} = \frac{321}{80}

So the first radical term becomes:

[32180]1/2\left[\frac{321}{80}\right]^{1/2}

Simplifying the Second Radical Term: Now simplify the second bracketed term:

9+1625+239 + \frac{16}{25} + \frac{2}{3}

To add these terms, we need a common denominator for the fractions. The LCM of 25 and 3 is 75. Convert the whole number and fractions to have this denominator:

9=9×7575=675759 = \frac{9 \times 75}{75} = \frac{675}{75}

1625=16×325×3=4875\frac{16}{25} = \frac{16 \times 3}{25 \times 3} = \frac{48}{75}

23=2×253×25=5075\frac{2}{3} = \frac{2 \times 25}{3 \times 25} = \frac{50}{75}

Now add the fractions:

67575+4875+5075=675+48+5075=77375\frac{675}{75} + \frac{48}{75} + \frac{50}{75} = \frac{675 + 48 + 50}{75} = \frac{773}{75}

So the second radical term becomes:

[77375]1/2\left[\frac{773}{75}\right]^{1/2}

Simplifying the Third Term: Finally, simplify the third term:

[(23)−2+1]\left[\left(\frac{2}{3}\right)^{-2} + 1\right]

Recall that a−b=1aba^{-b} = \frac{1}{a^b}. Therefore:

(23)−2=(32)2=3222=94\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{3^2}{2^2} = \frac{9}{4}

Now add 1:

94+1=94+44=134\frac{9}{4} + 1 = \frac{9}{4} + \frac{4}{4} = \frac{13}{4}

So the third term simplifies to:

134\frac{13}{4}

Combining the Radical Terms: Now, we multiply all the simplified radical terms together:

[32180]1/2â‹…[77375]1/2â‹…134\left[\frac{321}{80}\right]^{1/2} \cdot \left[\frac{773}{75}\right]^{1/2} \cdot \frac{13}{4}

Combine the square roots:

32180â‹…77375â‹…134\sqrt{\frac{321}{80} \cdot \frac{773}{75}} \cdot \frac{13}{4}

Multiply the fractions inside the square root:

321×77380×75⋅134=2481336000⋅134\sqrt{\frac{321 \times 773}{80 \times 75}} \cdot \frac{13}{4} = \sqrt{\frac{248133}{6000}} \cdot \frac{13}{4}

Simplify the fraction inside the square root by dividing both numerator and denominator by 3:

827112000â‹…134\sqrt{\frac{82711}{2000}} \cdot \frac{13}{4}

This square root does not simplify to a neat integer or fraction, so we leave it as is. Thus, the simplified radical expression is:

827112000â‹…134\sqrt{\frac{82711}{2000}} \cdot \frac{13}{4}

Final Simplified Expression

Now, let's combine all the simplified parts:

Numerical Exponent Part: 23n+42^{3n+4}

Variable Exponent Part: x6n−3x^{6n-3}

Radical Terms Part: 827112000â‹…134\sqrt{\frac{82711}{2000}} \cdot \frac{13}{4}

Thus, the final simplified expression is:

23n+4⋅x6n−3⋅827112000⋅1342^{3n+4} \cdot x^{6n-3} \cdot \sqrt{\frac{82711}{2000}} \cdot \frac{13}{4}

Conclusion

By systematically breaking down the original complex algebraic expression, we simplified it step by step. We addressed numerical exponents, variable exponents, and radical terms individually, and then combined the results. This approach ensures clarity and accuracy in simplifying complex mathematical expressions.

This detailed walkthrough should provide a comprehensive understanding of the simplification process, making it easier to tackle similar problems in the future. Remember to always break down complex problems into smaller, manageable steps for the best results.