Multiplication Laws Of Exponents

Master the laws for multiplying powers with the same base, raising a power to a power, and multiplying powers with the same exponent.

7 lessons

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Concept

Multiplying Same Bases

Introduce the product law for the same base.

Have you ever wondered how fast things grow when they multiply repeatedly?

When working with exponential growth, we often need to multiply numbers that share the same base. Let's break down how this works using a simple, observable pattern.

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Multiplication Laws of Exponents Visual

Visual grouping of exponential terms to show why exponents add.

rich vibrant illustration, Kurzgesagt-inspired, bold shapes with subtle texturing, saturated but harmonious color palett…

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Law 1: Product of Powers

Formula for $n^a \times n^b.$

n^{a} \times n^{b} = n^{a + b}

Where $a$ and $b$ are counting numbers.

Example Substitution:

3^{4} \times 3^{3} = 3^{4 + 3} = 3^{7}

Law 2: Power of a Power

(n^{a})^{b} = (n^{b})^{a} = n^{a \times b}

Multiply exponents when raising a power to a power.

Law 3: Same Exponents

m^{a} \times n^{a} = (m \times n)^{a}

Example:

3^{4} \times 2^{4} = (3 \times 2)^{4} = 6^{4}

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Example

Power of a Power

Introduce the power of a power law.

Problem. Simplify the expression $(4^3)^2$ and write it as a single power of $4$ .

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Law 2 and 3: Power of Power & Same Exponents

Formulas for (n^a)^b and m^a * n^a.

Law 2: Power of a Power

(n^{a})^{b} = (n^{b})^{a} = n^{a \times b}

Multiply exponents when raising a power to a power.

Law 3: Same Exponents

m^{a} \times n^{a} = (m \times n)^{a}

Example:

3^{4} \times 2^{4} = (3 \times 2)^{4} = 6^{4}

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reading

Simplify Expressions

Faded practice for product laws.

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Let us simplify the expression $8^6$ by rewriting it as a power of $2$ . First, we need to express the base $8$ as a prime factor in exponential form. We know that $8 = 2 \times 2 \times 2$ , which means $8$ can be written as $2$ to the power of . Substituting this back into our original expression gives us $8^6 = (2^3)^6$ . Next, we apply the general rule for a power of a power, which states that $(n^a)^b = n^{a \times b}$ . By multiplying the inner and outer exponents together, we find that the final simplified expression is $2^{ ␟blank2␟ }$ .

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quiz

Combining Rules

MCQ testing mixed product rules.

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Simplify the expression: $m^5 \times n^{12} \times (mn)^9$ .

← previousBasics Of Exponential Notation next →Division, Zero, And Negative Exponents

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Law 1: Product of Powers

Formula for $n^a \times n^b.$

n^{a} \times n^{b} = n^{a + b}

Where $a$ and $b$ are counting numbers.

Example Substitution:

3^{4} \times 3^{3} = 3^{4 + 3} = 3^{7}

Law 2: Power of a Power

(n^{a})^{b} = (n^{b})^{a} = n^{a \times b}

Multiply exponents when raising a power to a power.

Law 3: Same Exponents

m^{a} \times n^{a} = (m \times n)^{a}

Example:

3^{4} \times 2^{4} = (3 \times 2)^{4} = 6^{4}

Example

Power of a Power

Introduce the power of a power law.

Problem. Simplify the expression $(4^3)^2$ and write it as a single power of $4$ .

1 / 4

reading

Simplify Expressions

Faded practice for product laws.

0 of 2 blanks filled

quiz

Combining Rules

MCQ testing mixed product rules.

1 / 5

Simplify the expression: $m^5 \times n^{12} \times (mn)^9$ .