Establish and apply exponent laws

Power

\(\displaystyle 3^5\)

What does the exponent tell us in the power shown?

Keep the requested form separate from the value

The base is two and the exponent is four. The expanded form shows all four factors, and the evaluated value is sixteen

When asked for one power of the original base, write the base and exponent rather than only its evaluated value

An exponent of one means one factor

This section asks you to include the exponent even when it is one, so the factor count stays explicit

Repeated product

\(\displaystyle 5\times5\times5\times5=5^{\square}\)

Enter the exponent that writes the repeated product as one power of five

Count factors, not multiplication signs

Join repeated factors of the same base

Joining three factors and two more factors gives five factors of two

The base stays unchanged. Add the exponents because you are adding factor counts

Using b for the common numerical base and m and n for the exponents gives the product law

Retain the original numerical base

Write the expression as one power of the original numerical base. Include the exponent even when it is one

Multiplying powers of the same complete base joins their factor counts

Keep the original base. The index counts its remaining repeated factors

Evaluate the resulting power as a check, without replacing the requested single-power answer

Add the two factor counts

Sum

\(\displaystyle 2^2+2^3\)

Which statement about the sum is correct?

Product

\(\displaystyle 2^3\times3^2\)

Which conclusion is justified for this product?

Cancel common non-zero factors

Cancel two common factors of four. Three factors remain in the numerator

For a non-zero common numerical base, subtract the denominator exponent from the numerator exponent

In this section the numerator exponent is at least the denominator exponent, so the final exponent is not negative

Subtract the denominator factor count

Write the expression as one power of the original numerical base. Include the exponent even when it is one

The base is non-zero, so equal factors in the numerator and denominator can be cancelled

Cancel 2 common factors from 4 numerator factors

Keep the original base. The index counts its remaining repeated factors

Evaluate the resulting power as a check, without replacing the requested single-power answer

Retain the factors left after cancellation

Show the remaining single factor as a power

Explain why the zero power is one

A non-zero number divided by itself equals one

The quotient law gives the same expression an exponent of three minus three

For consistency, the zero power of this non-zero base must be one

The non-zero condition matters because the explanation uses division by a power of the base

Choose a base, then decrease n one step at a time to zero. Compare the two value columns

What consistent rule reaches exponent zero?

A negative non-zero base also has zero power one

Evaluate the expression as an integer

The stated base \(\left(-4\right)\) is non-zero

A non-zero number divided by itself is one

Subtracting the equal exponents gives zero, so the zero power of this non-zero base must equal one

Apply the zero law to a non-zero fraction

Which statement follows from the zero-exponent law used here?

Count repeated groups of factors

There are three groups, each containing two factors of three

Multiply the inner and outer exponents when a power is raised to a power

The brackets matter. A product of two powers adds factor counts, while a power of a power repeats the whole group

Two groups of two factors

Write as one power with the original base, without evaluating

The outer exponent gives 2 groups, each containing 2 copies of the complete base

Multiply the groups below, retaining every factor and every negative sign

Count the factors by multiplying the inner and outer exponents

An exact evaluation checks the result, but the requested answer remains a single power

Keep the complete original base in the final power

Multiply the group count by the factors per group

One group keeps the original power

Treat a signed number or fraction as one complete base

The same repeated-factor reasoning works for negative and fractional numerical bases. Brackets keep the complete base together

A negative sign outside a power is not automatically part of its base

Use the original complete numerical base when writing a single-power answer

Keep a negative base inside brackets

Keep the fractional base unchanged

A zero exponent contributes a factor of one

The zero-exponent rule and product law agree. A zero exponent does not make the whole product zero

Combine a zero power with a varying positive power

Combine numerator factors before cancelling denominator factors

The numerator product adds two and three. The denominator removes one factor

Write each law in a separate step so addition and subtraction of indices are not confused

If the first power is itself raised to a power, count that group before joining the numerator factors

Use a product law followed by a quotient law

Write the expression as one power of the original numerical base. Include the exponent even when it is one

Multiplying powers of the same complete base joins their factor counts

The base is non-zero, so equal factors in the numerator and denominator can be cancelled

Cancel 3 common factors from 6 numerator factors

Keep the original base. The index counts its remaining repeated factors

Evaluate the resulting power as a check, without replacing the requested single-power answer

Add numerator indices, then subtract the denominator index

Apply three laws in order

Write the expression as one power of the original numerical base. Include the exponent even when it is one

The first power is repeated 2 times, so multiply its inner and outer indices

Multiplying powers of the same complete base joins their factor counts

The base is non-zero, so equal factors in the numerator and denominator can be cancelled

Cancel 5 common factors from 7 numerator factors

Keep the original base. The index counts its remaining repeated factors

Evaluate the resulting power as a check, without replacing the requested single-power answer

Count nested factors before the other powers

Use numerical powers to compare cube volumes

A cube with side length s has volume s times s times s, or s cubed. The side lengths in a ratio must use the same length unit

For cubes with side lengths three centimetres and nine centimetres, compare the larger volume with the smaller

The larger cube has twenty-seven times the smaller volume, not three times. The volume ratio is dimensionless because the same cubic unit cancels

Construct a volume ratio as one power

Side lengths

\(\displaystyle 2\,\mathrm{cm}\quad\text{and}\quad4\,\mathrm{cm}\)

Two cubes have side lengths two centimetres and four centimetres. How many times the smaller volume is the larger volume?

Compare cubed side lengths rather than only the side lengths

Investigate systematically without treating examples as proof

This independent investigation is not automatically marked. Use a calculator or spreadsheet after predicting the values

Fix the base at two and the first exponent at three. Vary the second exponent from zero through four, changing only that exponent

Record both values for each exponent and any equality you observe. Repeat using a negative non-zero base, entering brackets carefully

Test a quotient with equal positive exponents and a power raised to exponent one. Explain the results using factors, not only a matching display

A mismatched display may come from an incorrect expression entry. A list of matches illustrates a law but does not prove it for every permitted input

Connect the digital pattern to factor counting

The second expressions give these same values because each increase of n contributes one more factor of two

The parentheses keep negative two as the complete base in both forms

Cancellation explains the equal-power quotient, and one repeated group explains the outer exponent of one

State the law and keep its conditions

For powers of the same complete numerical base, multiplication joins factor counts and division removes common non-zero factors

A power raised to a power repeats its factor group, so multiply the exponents

These tasks use whole-number exponents and keep the resulting exponents nonnegative. Negative and fractional exponents are not introduced here

Use one law at a time in a combined calculation and distinguish a single power from its evaluated value

The guided activities and fixed zero checks do not establish independent mastery of every exponent-law application