Knowledge Check: Chapters 7–12

Simplify: $\tfrac{4a^2b}{12ab^2}$.

Perform the indicated operation and simplify: $\tfrac{4x}{x+2} \cdot \tfrac{x^2+5x+6}{12x^2}$.

Find $R(x)$ given $R(x)=f(x)\cdot g(x)$.

Perform the indicated operation and simplify: $\tfrac{6x^2-x+20}{x^2-81}-\tfrac{5x^2+11x-7}{x^2-81}$.

Perform the indicated operation and simplify: $\tfrac{2n^2+8n-1}{n^2-1}-\tfrac{n^2-7n-1}{1-n^2}$.

Perform the indicated operation and simplify: $\tfrac{\tfrac{1}{m}-\tfrac{1}{n}}{\tfrac{1}{n}+\tfrac{1}{m}}$.

Solve the equation: $\tfrac{1}{z-5}+\tfrac{1}{z+5}=\tfrac{1}{z^2-25}$.

If $y$ varies inversely with the square of $x$ and $x=3$ when $y=9$, find $y$ when $x=4$.

Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver’s dad working alone to split the logs?

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The map distances are 2.4 inches from Dayton to Cincinnati, 3.2 inches from Dayton to Columbus, and 5.3 inches from Cincinnati to Columbus. The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?

Simplify using absolute values as necessary: $\sqrt[3]{125x^9}$.

Simplify using absolute values as necessary: $\sqrt[3]{72x^8y^4}$.

Simplify: $256^{-\tfrac{1}{4}}$.

Simplify: $-49^{\tfrac{3}{2}}$.

Simplify: $\tfrac{x^{-\tfrac{1}{4}}\cdot x^{\tfrac{5}{4}}}{x^{-\tfrac{3}{4}}}$.

Simplify: $\sqrt{48x^5}-\sqrt{75x^5}$.

Simplify: $2\sqrt{12x^5}\cdot 3\sqrt{6x^3}$.

Simplify: $(4-3\sqrt{3})(5+2\sqrt{3})$.

Simplify: $\tfrac{\sqrt{245xy^{-4}}}{\sqrt{45x^{-4}y^3}}$.

Simplify: $\tfrac{3}{2+\sqrt{3}}$.

Solve: $\sqrt{x+5}+1=x$.

Multiply: $-4i(-2-3i)$.

Simplify: $i^{172}$.

Use the Square Root Property to solve $3(w+5)^2=27$. Enter the smaller solution.

Use the Square Root Property to solve $3(w+5)^2=27$. Enter the larger solution.

Solve by completing the square: $d^2+14d=-13$. Enter the smaller solution.

Solve by completing the square: $d^2+14d=-13$. Enter the larger solution.

Use the Quadratic Formula to solve $2m^2-5m+3=0$. Enter the smaller solution.

Use the Quadratic Formula to solve $2m^2-5m+3=0$. Enter the larger solution.

Solve the quadratic equation using any method: $\tfrac{9}{4}y^2-3y+1=0$.

Solve $y^{\tfrac{2}{3}}+2y^{\tfrac{1}{3}}-3=0$. Enter the smaller solution.

Solve $y^{\tfrac{2}{3}}+2y^{\tfrac{1}{3}}-3=0$. Enter the larger solution.

The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. Round to the nearest tenth.

Find the equation of the axis of symmetry.

Find the vertex.

Find the $y$-intercept.

Find the $x$-intercept.

Find the maximum value.

Find $(f\circ g)(x)$.

Find $(g\circ f)(x)$.

Find $(f\cdot g)(x)$.

Find the inverse of the function $f(x)=x^5-9$.

Solve the equation $2^{2x-4}=64$.

Find the amount when interest is compounded quarterly.

Find the amount when interest is compounded monthly.

Find the amount when interest is compounded continuously.

Convert the equation from logarithmic to exponential form: $3=\log_7 343$.

Evaluate $\log_{11}1$.

What is the decibel level of a small fan with intensity $10^{-8}$ watts per square inch?

Use properties of logarithms to write $\log_5 25ab$ as a sum of logarithms, simplifying if possible.

Use properties of logarithms to write $\log_2\sqrt[4]{\tfrac{5x^3}{16y^2z^7}}$ as a sum of logarithms, simplifying if possible.

Use the properties of logarithms to condense $\tfrac{1}{6}\log x-3\log(x+5)$, simplifying if possible.

Solve for $x$: $\log_7(x+2)+\log_7(x-3)=\log_7 24$.

Solve $5e^{x-4}=40$. Give the exact answer.

Solve $5e^{x-4}=40$. Approximate the answer to three decimal places.

Researchers recorded that a certain bacteria population grew from 500 to 700 in 5 hours. At this rate of growth, how many bacteria will there be in 20 hours?

Find the distance between the points. Round to the nearest tenth as needed.

Find the midpoint of the line segment.

Write the standard form of the equation of the circle with radius 11 and center $(0,0)$.

Write the standard form of the equation of the circle with center $(-2,3)$ and a point on the circle $(2,-3)$.

Write $x^2+y^2+10x+6y+30=0$ in standard form.

Write $y=2x^2-4x-2$ in standard form.

A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 20 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, use the graph with vertices $(-45,0)$ and $(45,0)$ and a focus $(25,0)$ to write an equation for the elliptical orbit of the comet.

Write $9x^2-25y^2-36x-50y-214=0$ in standard form.

Solve the nonlinear system $x^2+9y^2=9$ and $2x^2-9y^2=18$. Enter the solution with the negative $x$-coordinate.

Solve the nonlinear system $x^2+9y^2=9$ and $2x^2-9y^2=18$. Enter the solution with the positive $x$-coordinate.

Write the first five terms of the sequence whose general term is $a_n=\tfrac{(n+2)!}{(n+3)!}$.

Expand the partial sum and find its value: $\sum_{i=1}^{4}(-4)^i$.

Write the first five terms of the arithmetic sequence with first term $a_1=-13$ and common difference $d=3$.

Find the twenty-third term of an arithmetic sequence whose seventh term is 11 and common difference is 3.

Find a formula for the general term of an arithmetic sequence whose seventh term is 11 and common difference is 3.

Find the sum of the first 25 terms of the arithmetic sequence $5,9,13,17,21,\ldots$.

Find the sum: $\sum_{i=1}^{40}(5i-21)$.

Find the common ratio.

In the geometric sequence whose first term and common ratio are $a_1=5$ and $r=4$, find $a_{11}$.

Find the sum of the first thirteen terms of the geometric sequence $2,-6,18,-54,162,-486,\ldots$.

Find the sum: $1-\tfrac{1}{5}+\tfrac{1}{25}-\tfrac{1}{125}+\tfrac{1}{625}-\tfrac{1}{3125}+\cdots$.

Dave just got his first full-time job after graduating from high school at age 18. He decided to invest $450 per month in an IRA (an annuity). The interest on the annuity is 6% which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-fifth birthday?

Evaluate the binomial coefficient $\binom{8}{1}$.

Evaluate the binomial coefficient $\binom{16}{16}$.

Evaluate the binomial coefficient $\binom{12}{0}$.

Evaluate the binomial coefficient $\binom{10}{6}$.