Which Expression Is Equivalent To B 4 ⋅ B 3 B^4 \cdot B^3 B 4 ⋅ B 3 ?A. ( B ⋅ B ⋅ B ⋅ B ) + ( B ⋅ B ⋅ B (B \cdot B \cdot B \cdot B) + (B \cdot B \cdot B ( B ⋅ B ⋅ B ⋅ B ) + ( B ⋅ B ⋅ B ]B. ( B + B + B + B ) ⋅ ( B + B + B (B + B + B + B) \cdot (B + B + B ( B + B + B + B ) ⋅ ( B + B + B ]C. ( B ⋅ B ⋅ B ⋅ B ) ⋅ ( B ⋅ B ⋅ B (B \cdot B \cdot B \cdot B) \cdot (B \cdot B \cdot B ( B ⋅ B ⋅ B ⋅ B ) ⋅ ( B ⋅ B ⋅ B ]D. $(B \cdot

Introduction

When working with exponential expressions, it's essential to understand how to simplify and manipulate them to find equivalent forms. In this article, we'll focus on the expression $B^4 \cdot B^3$ and explore which of the given options is equivalent to it.

Understanding Exponential Expressions

Before we dive into the solution, let's review the basics of exponential expressions. An exponential expression is a mathematical expression that represents a quantity raised to a power. In this case, we have $B^4$, which means $B$ raised to the power of 4, and $B^3$, which means $B$ raised to the power of 3.

The Product of Powers Property

One of the fundamental properties of exponential expressions is the product of powers property, which states that when multiplying two exponential expressions with the same base, we can add their exponents. In other words, $a^m \cdot a^n = a^{m+n}$.

Applying the Product of Powers Property

Now, let's apply the product of powers property to the expression $B^4 \cdot B^3$. Since both expressions have the same base $B$, we can add their exponents:

$$B^4 \cdot B^3 = B^{4+3} = B^7$$

Evaluating the Options

Now that we have simplified the expression $B^4 \cdot B^3$ to $B^7$, let's evaluate the given options:

A. $(B \cdot B \cdot B \cdot B) + (B \cdot B \cdot B)$

This option is incorrect because it represents the sum of two exponential expressions, not their product.

B. $(B + B + B + B) \cdot (B + B + B)$

This option is also incorrect because it represents the product of two sums, not the product of two exponential expressions.

C. $(B \cdot B \cdot B \cdot B) \cdot (B \cdot B \cdot B)$

This option is incorrect because it represents the product of two exponential expressions with different bases, not the same base.

D. $(B \cdot B \cdot B \cdot B) \cdot (B \cdot B \cdot B \cdot B)$

This option is correct because it represents the product of two exponential expressions with the same base, which can be simplified using the product of powers property.

Conclusion

In conclusion, the correct answer is option D, $(B \cdot B \cdot B \cdot B) \cdot (B \cdot B \cdot B \cdot B)$, which is equivalent to $B^7$. This example demonstrates the importance of understanding the product of powers property and how to apply it to simplify exponential expressions.

Additional Examples

To further reinforce your understanding of the product of powers property, let's consider a few more examples:

• $A^5 \cdot A^2 = A^{5+2} = A^7$
• $X^3 \cdot X^4 = X^{3+4} = X^7$
• $Y^2 \cdot Y^6 = Y^{2+6} = Y^8$

These examples illustrate how the product of powers property can be applied to simplify exponential expressions with the same base.

Tips and Tricks

When working with exponential expressions, keep the following tips and tricks in mind:

• Always check if the bases are the same before applying the product of powers property.
• Use the product of powers property to simplify exponential expressions with the same base.
• Be careful when multiplying exponential expressions with different bases, as the result will not be a simplified form.

By following these tips and tricks, you'll become more confident and proficient in simplifying exponential expressions and finding equivalent forms.

Common Mistakes to Avoid

When working with exponential expressions, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are a few common mistakes to watch out for:

• Failing to check if the bases are the same before applying the product of powers property.
• Incorrectly applying the product of powers property to exponential expressions with different bases.
• Not simplifying exponential expressions using the product of powers property.

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and reliable.

Conclusion

Q: What is the product of powers property?

A: The product of powers property is a fundamental concept in algebra that states that when multiplying two exponential expressions with the same base, we can add their exponents. In other words, $a^m \cdot a^n = a^{m+n}$.

Q: How do I apply the product of powers property?

A: To apply the product of powers property, you need to check if the bases are the same. If they are, you can add their exponents. For example, $B^4 \cdot B^3 = B^{4+3} = B^7$.

Q: What if the bases are different?

A: If the bases are different, you cannot apply the product of powers property. Instead, you need to multiply the expressions as they are. For example, $B^4 \cdot X^3 = B^4 \cdot X^3$.

Q: Can I simplify exponential expressions with the same base using the product of powers property?

A: Yes, you can simplify exponential expressions with the same base using the product of powers property. For example, $B^5 \cdot B^2 = B^{5+2} = B^7$.

Q: What if I have an exponential expression with a negative exponent?

A: If you have an exponential expression with a negative exponent, you can rewrite it as a fraction. For example, $B^{-3} = \frac{1}{B^3}$.

Q: Can I apply the product of powers property to exponential expressions with negative exponents?

A: Yes, you can apply the product of powers property to exponential expressions with negative exponents. For example, $B^{-3} \cdot B^2 = B^{-3+2} = B^{-1} = \frac{1}{B}$.

Q: What if I have an exponential expression with a zero exponent?

A: If you have an exponential expression with a zero exponent, the result is always 1. For example, $B^0 = 1$.

Q: Can I apply the product of powers property to exponential expressions with zero exponents?

A: Yes, you can apply the product of powers property to exponential expressions with zero exponents. For example, $B^0 \cdot B^3 = B^{0+3} = B^3$.

Q: What if I have an exponential expression with a fractional exponent?

A: If you have an exponential expression with a fractional exponent, you can rewrite it as a root. For example, $B^{\frac{1}{2}} = \sqrt{B}$.

Q: Can I apply the product of powers property to exponential expressions with fractional exponents?

A: Yes, you can apply the product of powers property to exponential expressions with fractional exponents. For example, $B^{\frac{1}{2}} \cdot B^{\frac{1}{3}} = B^{\frac{1}{2}+\frac{1}{3}} = B^{\frac{5}{6}}$.

Conclusion

In conclusion, the product of powers property is a fundamental concept in algebra that allows us to simplify exponential expressions with the same base. By understanding and applying this property, you can find equivalent forms of exponential expressions and solve problems with confidence. Remember to always check if the bases are the same before applying the product of powers property, and be careful when multiplying exponential expressions with different bases. With practice and patience, you'll become proficient in simplifying exponential expressions and finding equivalent forms.