\( a_6 = 3 \times 2^{5} = 3 \times 32 = 96 \).

["# Breaking Down the Calculation: ( a_6 = 3 \ imes 2^5 = 96 ) Explained", "Mathematics often combines simple operations in ways that reveal powerful insights—especially when it comes to exponents and multiplication. One classic example is the expression ( a_6 = 3 \ imes 2^5 ), which simplifies to ( 96 ) cleanly and efficiently. In this SEO-optimized guide, we’ll unpack this calculation step by step, explain its significance in algebra and number theory, and explore how understanding such expressions strengthens foundational math skills.", "### What Does ( 2^5 ) Mean?", "At the heart of ( a_6 = 3 \ imes 2^5 ) lies the expression ( 2^5 ), where ( 2 ) is the base and ( 5 ) is the exponent. Exponentiation refers to multiplying a number by itself repeatedly:", "( 2^5 = 2 \ imes 2 \ imes 2 \ imes 2 \ imes 2 )", "Calculating this step by step:\n- ( 2 \ imes 2 = 4 )\n- ( 4 \ imes 2 = 8 )\n- ( 8 \ imes 2 = 16 )\n- ( 16 \ imes 2 = 32 )", "So, ( 2^5 = 32 ).", "### Multiplying by 3: From ( 3 \ imes 32 ) to 96", "Now, plug ( 2^5 = 32 ) back into the original equation:", "( a_6 = 3 \ imes 32 )", "Multiplying these two numbers:\n- ( 3 \ imes 30 = 90 )\n- ( 3 \ imes 2 = 6 )\n- ( 90 + 6 = 96 )", "Thus, ( a_6 = 96 ).", "This transformation highlights how exponent operations scale quickly—multiplying ( 3 ) by a growing power of ( 2 ) rapidly builds toward a larger integer. It’s a foundational concept in exponential growth, relevant across science, finance, and computer science.", "### Why This Equation Matters: Exponents in Algebra", "Expressions like ( 3 \ imes 2^5 ) are more than number puzzles—they represent a core algebraic idea: variables can be expressed and simplified using exponents and multiplication. This form appears frequently in:\n- Exponential functions, modeling compound growth, population dynamics, or obsolescence of technology.\n- Financial mathematics, such as compound interest calculations.\n- Computer science, where binary systems rely heavily on powers of ( 2 ).", "Understanding how to decompose and compute such expressions builds fluency in algebraic reasoning.", "### Solving ( a_6 = ? ): Step-by-Step", "Want to master similar calculations? Follow this proven formula:", "1. Identify the exponent term: Notice that ( 2^5 ) represents exponentiation.\n - Tip: Watch for superscripts (small integers above a number)—these signal exponents.\n2. Calculate the exponent: Multiply the base by itself as many times as indicated by the exponent.\n - ( 2 \ imes 2 \ imes 2 \ imes 2 \ imes 2 = 32 )\n3. Multiply by the coefficient: Scale the exponent result by the multiplier (here, 3).\n - ( 3 \ imes 32 = 96 )", "This approach works universally—whether dealing with small numbers or scientific notation.", "### Real-World Applications of Exponential Expressions", "Scituate: Why does ( 3 \ imes 2^5 = 96 ) matter outside the classroom? Here are two contexts:", "🏢 Business Forecasting\nCompanies model revenue growth exponentially. If profits grow by ( 2^6 = 64 \ imes ) a base figure every year, a 3% increase scaled by this factor could lead to insights about scaling strategies.", "⚛️ Physics & Chemistry\nRadioactive decay follows exponential decay rules. Though often inverse (decay), faster-growing mathematical models (e.g., reaction rates) use ( a \ imes b^n ), where ( b > 1 ) represents positive growth—similar logic applies.", "### Tip: Expand Your Horizons with Powers of 2", "Since ( 2^5 = 32 ) and ( 2^{10} = 1024 ), recognizing patterns in powers of 2 accelerates mental math. Learn:\n- ( 2^4 = 16 ), ( 2^6 = 64 ), ( 2^7 = 128 ).\n- Mastering these connect to binary (base 2), crucial in computing.", "### Final Thoughts", "The calculation ( a_6 = 3 \ imes 2^5 = 96 ) is a gateway to deeper mathematical insight. By breaking it down—calculating exponents, applying multiplication, and connecting to real-world scaling—we uncover how math transforms complexity into clarity. Whether simplifying expressions or modeling growth, fluency with exponents empowers smarter problem-solving across disciplines.", "Keywords: ( 2^5 = 32 ), ( 3 \ imes 2^5 = 96 ), exponentiation, algebraic simplification, exponential growth, math fundamentals, scaling factors, intermediate math.", "Stay curious—every small calculation reveals big connections!"]