Rewrite 8 as 23 and apply the exponent rule a^m * aⁿ = a^m+n in the numerator.

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2.1 Simplify fully, without using a calculator: $\frac{2^{n+1} - 8 \cdot 2^{n-3}}{2^{n-2}}$

Rewrite 8 as $2^3$ and apply the exponent rule $a^m \cdot a^n = a^{m+n}$ in the numerator.
$\frac{2^{n+1} - 2^3 \cdot 2^{n-3}}{2^{n-2}} = \frac{2^{n+1} - 2^{3+(n-3)}}{2^{n-2}}$ $= \frac{2^{n+1} - 2^n}{2^{n-2}}$
Factor out the common term $2^n$ from the numerator.
$\frac{2^n(2^1 - 1)}{2^{n-2}} = \frac{2^n(2 - 1)}{2^{n-2}}$ $= \frac{2^n \cdot 1}{2^{n-2}} = \frac{2^n}{2^{n-2}}$
Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .
$2^{n - (n-2)} = 2^{n - n + 2}$ $= 2^2$
Calculate the final value.
$2^2 = 4$
The simplified expression is $\boxed{\text{4}}$ .

2.2 Solve for $x$ :

2.2.1 $\sqrt[x]{27} = 2187$
Rewrite the radical as a fractional exponent and express both sides with base 3.
$27^{1/x} = 2187$
Since $27 = 3^3$ and $2187 = 3^7$ :
$(3^3)^{1/x} = 3^7$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$3^{3/x} = 3^7$
Equate the exponents.
$\frac{3}{x} = 7$
Solve for $x$ .
$3 = 7x$ $x = \frac{3}{7}$
The solution is $\boxed{x = \frac{3}{7}}$ .

2.2.2 $4^x - 16 = 6 \cdot 2^x$
Rewrite $4^x$ as $(2^2)^x = (2^x)^2$ .
$(2^x)^2 - 16 = 6 \cdot 2^x$
Let $y = 2^x$ to form a quadratic equation.
$y^2 - 16 = 6y$
Rearrange into standard quadratic form:
$y^2 - 6y - 16 = 0$
Factor the quadratic equation.
$(y - 8)(y + 2) = 0$
Solve for $y$ .
$y - 8 = 0 \implies y = 8$ $y + 2 = 0 \implies y = -2$
Substitute back $2^x$ for $y$ and solve for $x$ .
For $y = 8$ :
$2^x = 8$ $2^x = 2^3$ $x = 3$
For $y = -2$ :
$2^x = -2$
There is no real solution for $x$ since $2^x$ must always be positive.

The solution is $\boxed{x = 3}$ .

2.3 Given that $x = \sqrt{3} - 2$ , simplify $\frac{x^2+1}{x^2-5}$ without using a calculator. (Give your answer in simplest surd form.)
Calculate $x^2$ .
$x^2 = (\sqrt{3} - 2)^2$ $x^2 = (\sqrt{3})^2 - 2(\sqrt{3})(2) + (-2)^2$ $x^2 = 3 - 4\sqrt{3} + 4$ $x^2 = 7 - 4\sqrt{3}$
Substitute $x^2$ into the numerator $x^2+1$ .
$x^2+1 = (7 - 4\sqrt{3}) + 1$ $x^2+1 = 8 - 4\sqrt{3}$
Substitute $x^2$ into the denominator $x^2-5$ .
$x^2-5 = (7 - 4\sqrt{3}) - 5$ $x^2-5 = 2 - 4\sqrt{3}$
Form the fraction and simplify by factoring out common terms.
$\frac{x^2+1}{x^2-5} = \frac{8 - 4\sqrt{3}}{2 - 4\sqrt{3}}$ $= \frac{4(2 - \sqrt{3})}{2(1 - 2\sqrt{3})}$ $= \frac{2(2 - \sqrt{3})}{1 - 2\sqrt{3}}$
Rationalize the denominator by multiplying the numerator and denominator by the conjugate of $1 - 2\sqrt{3}$ , which is $1 + 2\sqrt{3}$ .
$\frac{2(2 - \sqrt{3})}{1 - 2\sqrt{3}} \times \frac{1 + 2\sqrt{3}}{1 + 2\sqrt{3}}$
Expand the numerator.
$2(2 - \sqrt{3})(1 + 2\sqrt{3}) = 2(2 \cdot 1 + 2 \cdot 2\sqrt{3} - \sqrt{3} \cdot 1 - \sqrt{3} \cdot 2\sqrt{3})$ $= 2(2 + 4\sqrt{3} - \sqrt{3} - 2 \cdot 3)$ $= 2(2 + 3\sqrt{3} - 6)$ $= 2(-4 + 3\sqrt{3})$ $= -8 + 6\sqrt{3}$
Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .
$(1 - 2\sqrt{3})(1 + 2\sqrt{3}) = 1^2 - (2\sqrt{3})^2$ $= 1 - (4 \cdot 3)$ $= 1 - 12$ $= -11$
Combine the simplified numerator and denominator.
$\frac{-8 + 6\sqrt{3}}{-11} = \frac{8 - 6\sqrt{3}}{11}$
The simplified expression is $\boxed{\frac{8 - 6\sqrt{3}}{11}}$ .

Rewrite 8 as 23 and apply the exponent rule a^m * aⁿ = a^m+n in the numerator.

Step-by-step solution

Rewrite 8 as $2^3$ and apply the exponent rule $a^m \cdot a^n = a^{m+n}$ in the numerator.

Factor out the common term $2^n$ from the numerator.

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Calculate the final value.

Rewrite the radical as a fractional exponent and express both sides with base 3.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Equate the exponents.

Solve for $x$ .

Rewrite $4^x$ as $(2^2)^x = (2^x)^2$ .

Let $y = 2^x$ to form a quadratic equation.

Factor the quadratic equation.

Solve for $y$ .

Substitute back $2^x$ for $y$ and solve for $x$ .

Calculate $x^2$ .

Substitute $x^2$ into the numerator $x^2+1$ .

Substitute $x^2$ into the denominator $x^2-5$ .

Form the fraction and simplify by factoring out common terms.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of $1 - 2\sqrt{3}$ , which is $1 + 2\sqrt{3}$ .

Expand the numerator.

Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .

Combine the simplified numerator and denominator.

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More Mathematics Questions

Rewrite 8 as 23 and apply the exponent rule a^m * aⁿ = a^m+n in the numerator.

Step-by-step solution

Rewrite 8 as 232^323 and apply the exponent rule am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n in the numerator.

Factor out the common term 2n2^n2n from the numerator.

Apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Calculate the final value.

Rewrite the radical as a fractional exponent and express both sides with base 3.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Equate the exponents.

Solve for xxx.

Rewrite 4x4^x4x as (22)x=(2x)2(2^2)^x = (2^x)^2(22)x=(2x)2.

Let y=2xy = 2^xy=2x to form a quadratic equation.

Factor the quadratic equation.

Solve for yyy.

Substitute back 2x2^x2x for yyy and solve for xxx.

Calculate x2x^2x2.

Substitute x2x^2x2 into the numerator x2+1x^2+1x2+1.

Substitute x2x^2x2 into the denominator x2−5x^2-5x2−5.

Form the fraction and simplify by factoring out common terms.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of 1−231 - 2\sqrt{3}1−23​, which is 1+231 + 2\sqrt{3}1+23​.

Expand the numerator.

Expand the denominator using the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2.

Combine the simplified numerator and denominator.

Still stuck on this one?

More Mathematics Questions

Rewrite 8 as $2^3$ and apply the exponent rule $a^m \cdot a^n = a^{m+n}$ in the numerator.

Factor out the common term $2^n$ from the numerator.

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Solve for $x$ .

Rewrite $4^x$ as $(2^2)^x = (2^x)^2$ .

Let $y = 2^x$ to form a quadratic equation.

Solve for $y$ .

Substitute back $2^x$ for $y$ and solve for $x$ .

Calculate $x^2$ .

Substitute $x^2$ into the numerator $x^2+1$ .

Substitute $x^2$ into the denominator $x^2-5$ .

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of $1 - 2\sqrt{3}$ , which is $1 + 2\sqrt{3}$ .

Expand the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ .