Simplify the following expressions:

1. \(\left(4^5\right)^6\)

2. \(\left(x^3\right)^5\)

3. \(\left(x^4\right)^3\left(x^2\right)^5\)

Simplify the following the expressions:

1. \((3y)^2\)

2. \(\left(-\displaystyle\frac{1}{4}x^2y^3\right)^2\)

3. \(\left(x^2y^5\right)^3\left(x^4y\right)^2\)

Mini Lecture
Simplify.

1. \(\left(\displaystyle\frac{1}{2}\right)^4\)

2. \(x^4\cdot x^5\)

3. \(\left(x^2\right)^5\)

4. \((4xyz)^3\)

5. \(\left(3x^2\right)\left(2x^3\right)\left(5x^4\right)\)

6. Write in scientific notation: \(43,200\)

7. Write in expanded form: \(2.49\times 10^3\)

Write each expression with a positive exponent and then simplify:

1. \(2^{-3}\)

2. \((-5)^{-2}\)

3. \(-5^{-2}\)

4. \(3x^{-6}\)

5. \((3x)^{-6}\)

Write each number in scientific notation.

1. \(36\times 10^{-4}\)

2. \(0.36\times 10^5\)

Mini Lecture
Simplify.

1. \(3^{-2}\)

2. \(5^{-3}\)

3. \(\displaystyle\frac{x^{10}}{x^4}\)

4. \(\displaystyle\frac{5^1}{5^3}\)

5. \(\displaystyle\frac{2^5}{2^{-3}}\)

6. \(\displaystyle\frac{\left(a^3\right)^2\left(a^4\right)^5}{\left(a^5\right)^2}\)

7. Write in expanded form: \(4.23\times 10^{-3}\)

8. Write in scientific notation: \(0.000357\)

State the coefficient and degree of the following monomial: \(8x^5\)

State the coefficient and degree of the following monomial: \(-\dfrac{1}{2}x\)

State the coefficient and degree of the following monomial: \(14x^4y^3\)

State the coefficient and degree of the following monomial: \(-3\)

Simplify each expression.

1. \((2y)^3(3y)^2\)

2. \(\dfrac{(6xy^4)^2}{(2x^3y^2)^3}\)

Simplify: \(\dfrac{(6.8\times 10^5)(3.9\times 10^{-7})}{1.7\times 10^{-4}}\)

A rectangular solid is twice as long as it is wide and one-half as high as it is wide. Write an expression for the volume.

Mini Lecture
Simplify.

1. \(\left(4x^2y\right)\left(3x^3y^3\right)\left(2xy^4\right)\)

2. \(\displaystyle\frac{21a^2b^3}{-7ab^5}\)

3. \(\displaystyle\frac{8.4\times 10^5}{2\times 10^2}\)

4. \(10x^3-8x^3+2x^3\)

Write \(2x-5+3x^4-x^2\) in standard form. Then state the degree of the polynomial and the leading coefficient.

Add \(x^2+3x\) and \(2x+6\).

Find the opposite of \(3x^2-5x+8\).

Mini Lecture
Indicate the degree.

1. \(4x^2-4x+1\)

2. \(90x\)

Combine.

3. \(\left(x^2+5x+6\right)+\left(x^2+3x+4\right)\)

4. \(\left(5a^2-a-6\right)-\left(-3a^2-2a+4\right)\)

5. Subtract \(3y^2+7y-15\) from \(11y^2+11y+11\)

Multiply: \((x^2-4x+2)(3x^2+x-5)\)

Multiply: \((3x-2)(4x^2+x+3)\)

A store selling cases for cell phones knows from past experience that it can sell \(x\) cases each day at a price \(p\) dollars per case, according to the equation \(x=800-100p\). Write a formula for the daily revenue that involves only the variable \(R\) and \(p\).

Mini Lecture
Multiply.

1. \(2x^2\left(3x^2-2x+1\right)\)

2. \((x+3)(x+4)\)

3. \((2x-5)(3x-2)\)

4. \((x+2)\left(x^2-2x+4\right)\)

Find the square of \((3x-2)\)

Multiply \((2x-3y)^2\) using the rule for binomial squares.

Multiply \(\left(5x^2-4\right)^2\) using the rule for binomial squares.

Multiply \((3x-1)(3x+1)\)

Multiply \((1+9a)(1-9a)\) using the rule for difference of two squares.

Multiply \((3a^2-7b^2)(3a^2+7b^2)\) using the rule for difference of two squares.

Write an expression in symbols for the sum of the squares of three consecutive even integers. Then simplify that expression.

Mini Lecture
Expand and multiply.

1. \((x+3)^2\)

2. \((3a+5b)^2\)

3. \((x-3)(x+3)\)

4. \((6-7x)(6+7x)\)

Enrichment
More powers of binomials, Pascal’s triangle, and the Fibonacci sequence.

Divide \(\left(15x^2y-21xy^2\right)\div (-3xy)\)

Divide: \(\dfrac{24x^3y^2+16x^2y^2-4xy^3}{8x^2y}\)

Divide \(27\overline{\smash{)}3,962}\)

Divide: \(\dfrac{6x^3-11x^2-14x+3}{2x^2-5x-1}\)

Divide: \(\dfrac{6x^3+2x^2-x}{2x^2-3}\)

Mini Lecture
Divide.

1. \(\displaystyle\frac{6x+8y}{2}\)

2. \(\displaystyle\frac{x^2y-x^3y^2}{-x^2y}\)

3. \(\displaystyle\frac{16x^5+8x^2+12x}{12x^3}\)

Simplify: \((-2)^5\)

Simplify: \(\left(\displaystyle\frac{2}{3}\right)^3\)

Simplify: \(\left(4x^2\right)^2\left(2x^3\right)^3\)

Simplify with positive exponents only. \[4^{-2}\]

Simplify with positive exponents only. \[\left(4a^5b^3\right)^0\]

Simplify with positive exponents only. \[\displaystyle\frac{x^{-4}}{x^{-7}}\]

Simplify with positive exponents only. \[\displaystyle\frac{\left(x^{-3}\right)^2\left(x^{-5}\right)^{-3}}{\left(x^{-3}\right)^{-4}}\]

Write \(0.04307\) in scientific notation.

Write \(7.63\times 10^6\) in expanded form.

Simplify with positive exponents only. \[\displaystyle\frac{17x^2y^5z^3}{51x^4y^2z}\]

Simplify with positive exponents only. \[\displaystyle\frac{\left(3a^3b\right)\left(4a^2b^5\right)}{24a^2b^4}\]

Simplify with positive exponents only. \[\displaystyle\frac{28x^4}{4x}+\displaystyle\frac{30x^7}{6x^4}\]

Simplify with positive exponents only. \[\displaystyle\frac{\left(1.1\times 10^5\right)\left(3\times 10^{-2}\right)}{4.4\times 10^{-5}}\]

Simplify: \(\;9x^2-2x+7x+4\)

Simplify: \(\;\left(4x^2+5x-6\right)-\left(2x^2-x-4\right)\)

Subtract \(2x+7\) from \(7x+3\).

Find the value of \(3a^2+4a+6\) when \(a\) is \(-3\).

Multiply: \(\;3x^2\left(5x^2-2x+4\right)\)

Multiply: \(\; \left(x+\displaystyle\frac{1}{4}\right)\left(x-\displaystyle\frac{1}{3}\right)\)

Multiply: \(\;(2x-3)(5x+6)\)

Multiply: \(\; (x+4)(x^2-4x+16)\)

Multiply: \(\;(x-6)^2\)

Multiply: \(\;(2a+4b)^2\)

Multiply: \(\;(3x-6)(3x+6)\)

Multiply: \(\left(x^2-4\right)\left(x^2+4\right)\)

Divide \(18x^3-36x^2+6x\;\) by \(\;6x\).

Divide: \(\;\displaystyle\frac{9x^2-6x-4}{3x-1}\)

Divide: \(\;\displaystyle\frac{4x^2-5x+6}{x-4}\)

Find the volume of a cube if the length of a side is \(3.2\) inches.

Find the volume of a rectangular solid if the length is three times the width, and the height is one third the width.