To solve the expression \( 2^3 \times 2^5 \) using the laws of indices, we need to apply one of the fundamental rules of exponents. Let's break this down step-by-step. ### Step 1: Understanding the Laws of Indices The law of indices that we will use states that when you multiply two powers with the same base, you add the exponents. This can be expressed mathematically as: \[ a^m \times a^n = a^{m+n} \] In our case, the base \( a \) is \( 2 \), and the exponents are \( 3 \) and \( 5 \). ### Step 2: Applying the Law Now, we can apply this law to our expression: \[ 2^3 \times 2^5 = 2^{3+5} \] ### Step 3: Performing the Addition Next, we add the exponents: \[ 3 + 5 = 8 \] So, we have: \[ 2^{3+5} = 2^8 \] ### Step 4: Final Result Thus, the expression \( 2^3 \times 2^5 \) simplifies to \( 2^8 \). ### Conclusion The correct answer is **A. \( 2^8 \)**. ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect or weaker: - **Option B: \( 2^{15} \)** This option suggests that we should add the exponents incorrectly as \( 3 + 5 + 7 \) or some other combination. However, the correct application of the law of indices only requires adding the exponents of the same base, which we did correctly as \( 3 + 5 = 8 \). - **Option C: \( 2^5 \)** This option implies that we only considered one of the exponents. It is incorrect because we must account for both exponents when multiplying powers with the same base. - **Option D: \( 8 \times 2^5 \)** This option is misleading. While \( 8 \) is equal to \( 2^3 \), the expression \( 8 \times 2^5 \) does not represent the same operation as \( 2^3 \times 2^5 \). Instead, it suggests a different multiplication that does not follow the laws of indices correctly. ### Summary - The law of indices states that when multiplying powers with the same base, you add the exponents. - For \( 2^3 \times 2^5 \), we add \( 3 + 5 = 8 \). - The final result is \( 2^8 \), which corresponds to option A. - Other options either misapply the laws of indices or misinterpret the multiplication of powers. This thorough understanding of the laws of indices will help you tackle similar problems in the future!