Simplify Powers: 15^2 × 15^4 Using Exponent Properties

Q: Why do we add exponents when multiplying powers with the same base?

Think of it this way: $ 15^2 $ means 15 × 15, and $ 15^4 $ means 15 × 15 × 15 × 15. When you multiply them together, you get six copies of 15 multiplied together, which is $ 15^6 $!

Q: What if the bases are different, like $ 3^2 \times 5^4 $ ?

You cannot combine different bases using the exponent addition rule. $ 3^2 \times 5^4 $ stays as $ 3^2 \times 5^4 $ or you calculate each part separately: 9 × 625 = 5625.

Q: How can I remember not to multiply the exponents?

Remember: Same base multiplication = ADD exponents. Only multiply exponents when raising a power to another power, like $ (15^2)^4 = 15^{2\times4} = 15^8 $.

Q: What if one of the exponents is negative?

The same rule applies! For example: $ 15^2 \times 15^{-3} = 15^{2+(-3)} = 15^{-1} $. Just add the exponents, even if one is negative.

Exponent Properties with Same Base Multiplication

Simplify the following equation:

$15^2\times15^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, multiplying exponents with the same base (A)

00:06 is equal to the same base raised to the sum of the exponents (N+M)

00:10 We will apply this formula to our exercise

00:13 We'll add up the exponents and raise them to this power

00:19 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$15^2\times15^4=$

Step-by-step solution

To solve the problem of simplifying $15^2 \times 15^4$ , we will use the rule for multiplying exponents with the same base.

According to the multiplication of powers rule: If $a$ is a real number and $m$ and $n$ are integers, then:

$a^m \times a^n = a^{m+n}$ .

Applying this rule to our problem, where the base $a$ is 15, and the exponents $m$ and $n$ are 2 and 4 respectively:

Step 1: Identify the base and exponents: $15^2$ and $15^4$ have the same base.
Step 2: Add the exponents: $2 + 4 = 6$ .
Step 3: Simplify the expression using the rule: $15^2 \times 15^4 = 15^{2+4} = 15^6$ .

Therefore, the simplified expression is $15^6$ .

Final Answer

$15^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $15^2 \times 15^4 = 15^{2+4} = 15^6$
Check: Verify by expanding: $15^2 = 225$ and $15^4 = 50625$ gives same result ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't calculate $15^2 \times 15^4$ as $15^{2\times4} = 15^8$ ! This mixes up the multiplication rule with the power rule and gives a much larger answer. Always add exponents when multiplying same bases: $a^m \times a^n = a^{m+n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying powers with the same base?

Think of it this way: $15^2$ means 15 × 15, and $15^4$ means 15 × 15 × 15 × 15. When you multiply them together, you get six copies of 15 multiplied together, which is $15^6$ !

What if the bases are different, like $3^2 \times 5^4$ ?

You cannot combine different bases using the exponent addition rule. $3^2 \times 5^4$ stays as $3^2 \times 5^4$ or you calculate each part separately: 9 × 625 = 5625.

Is $15^6$ the final answer, or should I calculate the actual number?

Unless specifically asked to calculate the numerical value, $15^6$ is the simplified form! It's much cleaner than writing out 11,390,625 and shows you understand exponent properties.

How can I remember not to multiply the exponents?

Remember: Same base multiplication = ADD exponents. Only multiply exponents when raising a power to another power, like $(15^2)^4 = 15^{2\times4} = 15^8$ .

What if one of the exponents is negative?

The same rule applies! For example: $15^2 \times 15^{-3} = 15^{2+(-3)} = 15^{-1}$ . Just add the exponents, even if one is negative.

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Simplify Powers: 15^2 × 15^4 Using Exponent Properties

Exponent Properties with Same Base Multiplication

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add exponents when multiplying powers with the same base?

What if the bases are different, like $3^2 \times 5^4$ ?

Is $15^6$ the final answer, or should I calculate the actual number?

How can I remember not to multiply the exponents?

What if one of the exponents is negative?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: 6^(3+2) Using Exponent Rules

Expand the Exponential Expression: 4^(4+6) Step-by-Step

Simplify 5×5^8: Solving an Exponential Multiplication Problem

Simplify 4^5 × 4^5: Multiplying Powers with Same Base

Simplify Powers: 15^2 × 15^4 Using Exponent Properties

Exponent Properties with Same Base Multiplication

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we add exponents when multiplying powers with the same base?

What if the bases are different, like 32×54 3^2 \times 5^4 32×54?

Is 156 15^6 156 the final answer, or should I calculate the actual number?

How can I remember not to multiply the exponents?

What if one of the exponents is negative?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Expand the Expression: 6^(3+2) Using Exponent Rules

Expand the Exponential Expression: 4^(4+6) Step-by-Step

Simplify 5×5^8: Solving an Exponential Multiplication Problem

Simplify 4^5 × 4^5: Multiplying Powers with Same Base

What if the bases are different, like $3^2 \times 5^4$ ?

Is $15^6$ the final answer, or should I calculate the actual number?